Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Introduction to Linear Transformation

Math 4A – Xianzhe Dai

UCSB

April 14 2014

Based on the 2013 Millett and Scharlemann Lectures

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System of Linear Equations:

a11x1 + a12x2 + . . .+ a1nxn = b1

a21x1 + a22x2 + . . .+ a2nxn = b2...

am1x1 + am2x2 + . . .+ amnxn = bm

can be written as matrix equation A~x = ~b.

New perspective: think of the LHS as a“function/map/transformation”, T (~x) = A~x . T maps/transformsa vector ~x to another vector A~x .

Two very nice properties it enjoys are

T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)T (c~u) = A(c~u) = cA~u = cT (~u)

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

System of Linear Equations:

a11x1 + a12x2 + . . .+ a1nxn = b1

a21x1 + a22x2 + . . .+ a2nxn = b2...

am1x1 + am2x2 + . . .+ amnxn = bm

can be written as matrix equation A~x = ~b.

New perspective: think of the LHS as a“function/map/transformation”, T (~x) = A~x . T maps/transformsa vector ~x to another vector A~x .

Two very nice properties it enjoys are

T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)T (c~u) = A(c~u) = cA~u = cT (~u)

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

System of Linear Equations:

a11x1 + a12x2 + . . .+ a1nxn = b1

a21x1 + a22x2 + . . .+ a2nxn = b2...

am1x1 + am2x2 + . . .+ amnxn = bm

can be written as matrix equation A~x = ~b.

New perspective: think of the LHS as a“function/map/transformation”, T (~x) = A~x . T maps/transformsa vector ~x to another vector A~x .

Two very nice properties it enjoys are

T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)T (c~u) = A(c~u) = cA~u = cT (~u)

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System of Linear Equations:

a11x1 + a12x2 + . . .+ a1nxn = b1

a21x1 + a22x2 + . . .+ a2nxn = b2...

am1x1 + am2x2 + . . .+ amnxn = bm

can be written as matrix equation A~x = ~b.

New perspective: think of the LHS as a“function/map/transformation”, T (~x) = A~x . T maps/transformsa vector ~x to another vector A~x .

Two very nice properties it enjoys are

T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)T (c~u) = A(c~u) = cA~u = cT (~u)

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Linear Transformations

Definition

A linear transformation is a function T : Rn → Rm with theseproperties:

For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).

Example: Let T : R1 → R1 be defined by T (x) = 5x .For any u, v ∈ R1,

T (u + v) = 5(u + v) = 5u + 5v = T (u) + T (v) and

for any c ∈ R, T (cu) = 5cu = c5u = cT (u).

So T is a linear transformation.

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Linear Transformations

Definition

A linear transformation is a function T : Rn → Rm with theseproperties:

For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).

Example: Let T : R1 → R1 be defined by T (x) = 5x .

For any u, v ∈ R1,

T (u + v) = 5(u + v) = 5u + 5v = T (u) + T (v) and

for any c ∈ R, T (cu) = 5cu = c5u = cT (u).

So T is a linear transformation.

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Linear Transformations

Definition

A linear transformation is a function T : Rn → Rm with theseproperties:

For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).

Example: Let T : R1 → R1 be defined by T (x) = 5x .For any u, v ∈ R1,

T (u + v) = 5(u + v) = 5u + 5v = T (u) + T (v) and

for any c ∈ R, T (cu) = 5cu = c5u = cT (u).

So T is a linear transformation.

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Linear Transformations

Definition

A linear transformation is a function T : Rn → Rm with theseproperties:

For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).

Example: Let T : R1 → R1 be defined by T (x) = 5x .For any u, v ∈ R1,

T (u + v) = 5(u + v) = 5u + 5v = T (u) + T (v) and

for any c ∈ R, T (cu) = 5cu = c5u = cT (u).

So T is a linear transformation.

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Some notes:

Most functions are not linear transformations.

For example:cos(x + y) 6= cos(x) + cos(y). Or (2x)2 6= 2(x2).

For any linear transformation T (~0) = ~0 (this rules outfunction f (x) = x + 5): Take c = 0, then

T (~0) = T (0 ·~0) = 0T (~0) = ~0.

The two conditions could be written as one: For any vectors~u, ~v ∈ Rn and real numbers a, b ∈ R,

T (a~u + b~v) = aT (~u) + bT (~v)

.

Example

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Some notes:

Most functions are not linear transformations. For example:cos(x + y) 6= cos(x) + cos(y). Or (2x)2 6= 2(x2).

For any linear transformation T (~0) = ~0 (this rules outfunction f (x) = x + 5): Take c = 0, then

T (~0) = T (0 ·~0) = 0T (~0) = ~0.

The two conditions could be written as one: For any vectors~u, ~v ∈ Rn and real numbers a, b ∈ R,

T (a~u + b~v) = aT (~u) + bT (~v)

.

Example

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Some notes:

Most functions are not linear transformations. For example:cos(x + y) 6= cos(x) + cos(y). Or (2x)2 6= 2(x2).

For any linear transformation T (~0) = ~0 (this rules outfunction f (x) = x + 5):

Take c = 0, then

T (~0) = T (0 ·~0) = 0T (~0) = ~0.

The two conditions could be written as one: For any vectors~u, ~v ∈ Rn and real numbers a, b ∈ R,

T (a~u + b~v) = aT (~u) + bT (~v)

.

Example

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Some notes:

Most functions are not linear transformations. For example:cos(x + y) 6= cos(x) + cos(y). Or (2x)2 6= 2(x2).

For any linear transformation T (~0) = ~0 (this rules outfunction f (x) = x + 5): Take c = 0, then

T (~0) = T (0 ·~0) = 0T (~0) = ~0.

The two conditions could be written as one: For any vectors~u, ~v ∈ Rn and real numbers a, b ∈ R,

T (a~u + b~v) = aT (~u) + bT (~v)

.

Example

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Some notes:

Most functions are not linear transformations. For example:cos(x + y) 6= cos(x) + cos(y). Or (2x)2 6= 2(x2).

For any linear transformation T (~0) = ~0 (this rules outfunction f (x) = x + 5): Take c = 0, then

T (~0) = T (0 ·~0) = 0T (~0) = ~0.

The two conditions could be written as one: For any vectors~u, ~v ∈ Rn and real numbers a, b ∈ R,

T (a~u + b~v) = aT (~u) + bT (~v)

.

Example

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Important example:

Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:

For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).

A linear transformation defined by a matrix is called a matrixtransformation.

Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).

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Important example:

Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:

For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).

A linear transformation defined by a matrix is called a matrixtransformation.

Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Important example:

Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:

For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).

A linear transformation defined by a matrix is called a matrixtransformation.

Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Important example:

Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:

For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).

A linear transformation defined by a matrix is called a matrixtransformation.

Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Important example:

Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:

For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).

A linear transformation defined by a matrix is called a matrixtransformation.

Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Important example:

Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:

For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).

A linear transformation defined by a matrix is called a matrixtransformation.

Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).

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Mona Lisa transformed6/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Matrix transformations are important and are also cool!

Example 1, a shear: Consider the matrix transformationT : R2 → R2 given by the 2× 2 matrix

A =

[1 3

20 1

]

For any horizontal vector ~x =

[x10

]

T (~x) = A~x =

[1 3

20 1

] [x10

]=

[x1 + 3

2 · 00 · x1 + 1 · 0

]=

[x10

]So T is the identity on horizontal vectors.

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Matrix transformations are important and are also cool!Example 1, a shear: Consider the matrix transformationT : R2 → R2 given by the 2× 2 matrix

A =

[1 3

20 1

]

For any horizontal vector ~x =

[x10

]

T (~x) = A~x =

[1 3

20 1

] [x10

]=

[x1 + 3

2 · 00 · x1 + 1 · 0

]=

[x10

]So T is the identity on horizontal vectors.

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Matrix transformations are important and are also cool!Example 1, a shear: Consider the matrix transformationT : R2 → R2 given by the 2× 2 matrix

A =

[1 3

20 1

]

For any horizontal vector ~x =

[x10

]

T (~x) = A~x =

[1 3

20 1

] [x10

]=

[x1 + 3

2 · 00 · x1 + 1 · 0

]=

[x10

]

So T is the identity on horizontal vectors.

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Matrix transformations are important and are also cool!Example 1, a shear: Consider the matrix transformationT : R2 → R2 given by the 2× 2 matrix

A =

[1 3

20 1

]

For any horizontal vector ~x =

[x10

]

T (~x) = A~x =

[1 3

20 1

] [x10

]=

[x1 + 3

2 · 00 · x1 + 1 · 0

]=

[x10

]So T is the identity on horizontal vectors.

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

For any vertical vector ~x =

[0x2

]

T (~x) = A~x =

[1 3

20 1

] [0x2

]=

[1 · 0 + 3

2 · x20 · 0 + 1 · x2

]=

[0x2

]+

[32x20

]So a vertical vector is pushed perfectly horizontally, a distance 3

2times its length:

(0, 1)

(2, 0)

(2, 1)(3/2, 1) (2+3/2, 1)x

1

2

x

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

For any vertical vector ~x =

[0x2

]

T (~x) = A~x =

[1 3

20 1

] [0x2

]=

[1 · 0 + 3

2 · x20 · 0 + 1 · x2

]=

[0x2

]+

[32x20

]So a vertical vector is pushed perfectly horizontally, a distance 3

2times its length:

(0, 1)

(2, 0)

(2, 1)(3/2, 1) (2+3/2, 1)x

1

2

x

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

For any vertical vector ~x =

[0x2

]

T (~x) = A~x =

[1 3

20 1

] [0x2

]=

[1 · 0 + 3

2 · x20 · 0 + 1 · x2

]=

[0x2

]+

[32x20

]

So a vertical vector is pushed perfectly horizontally, a distance 32

times its length:

(0, 1)

(2, 0)

(2, 1)(3/2, 1) (2+3/2, 1)x

1

2

x

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

For any vertical vector ~x =

[0x2

]

T (~x) = A~x =

[1 3

20 1

] [0x2

]=

[1 · 0 + 3

2 · x20 · 0 + 1 · x2

]=

[0x2

]+

[32x20

]So a vertical vector is pushed perfectly horizontally, a distance 3

2times its length:

(0, 1)

(2, 0)

(2, 1)(3/2, 1) (2+3/2, 1)x

1

2

x

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

For any vertical vector ~x =

[0x2

]

T (~x) = A~x =

[1 3

20 1

] [0x2

]=

[1 · 0 + 3

2 · x20 · 0 + 1 · x2

]=

[0x2

]+

[32x20

]So a vertical vector is pushed perfectly horizontally, a distance 3

2times its length:

(0, 1)

(2, 0)

(2, 1)(3/2, 1) (2+3/2, 1)x

1

2

x

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 2, scaling:Use

A =

[2 00 1

2

]

For any vector ~x =

[x1x2

]T (~x) = A~x =

[2 00 1

2

] [x1x2

]=

[2x1 + 0 · x2

0 · x1 + 12 · x2

]=

[2x1x22

]So T stretches horizontally and contracts vertically:

(0, 1)

(2, 0)

x

1

2

x

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 2, scaling:Use

A =

[2 00 1

2

]For any vector ~x =

[x1x2

]T (~x) = A~x =

[2 00 1

2

] [x1x2

]=

[2x1 + 0 · x2

0 · x1 + 12 · x2

]=

[2x1x22

]

So T stretches horizontally and contracts vertically:

(0, 1)

(2, 0)

x

1

2

x

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 2, scaling:Use

A =

[2 00 1

2

]For any vector ~x =

[x1x2

]T (~x) = A~x =

[2 00 1

2

] [x1x2

]=

[2x1 + 0 · x2

0 · x1 + 12 · x2

]=

[2x1x22

]So T stretches horizontally and contracts vertically:

(0, 1)

(2, 0)

x

1

2

x

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 2, scaling:Use

A =

[2 00 1

2

]For any vector ~x =

[x1x2

]T (~x) = A~x =

[2 00 1

2

] [x1x2

]=

[2x1 + 0 · x2

0 · x1 + 12 · x2

]=

[2x1x22

]So T stretches horizontally and contracts vertically:

(0, 1)

(2, 0)

x

1

2

x

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 3, reflection through a line:Use

A =

[0 11 0

]

T (

[x1x2

]) = A

[x1x2

]=

[0 11 0

] [x1x2

]=

[0x1 + 1 · x2

1 · x1 + 0 · x2

]=

[x2x1

]So T exchanges the two coordinates. Looks like reflection throughthe line x1 = x2:

(s, s)x

1

2

x

reflect

reflect

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Example 3, reflection through a line:Use

A =

[0 11 0

]T (

[x1x2

]) = A

[x1x2

]=

[0 11 0

] [x1x2

]=

[0x1 + 1 · x2

1 · x1 + 0 · x2

]=

[x2x1

]

So T exchanges the two coordinates. Looks like reflection throughthe line x1 = x2:

(s, s)x

1

2

x

reflect

reflect

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 3, reflection through a line:Use

A =

[0 11 0

]T (

[x1x2

]) = A

[x1x2

]=

[0 11 0

] [x1x2

]=

[0x1 + 1 · x2

1 · x1 + 0 · x2

]=

[x2x1

]So T exchanges the two coordinates.

Looks like reflection throughthe line x1 = x2:

(s, s)x

1

2

x

reflect

reflect

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 3, reflection through a line:Use

A =

[0 11 0

]T (

[x1x2

]) = A

[x1x2

]=

[0 11 0

] [x1x2

]=

[0x1 + 1 · x2

1 · x1 + 0 · x2

]=

[x2x1

]So T exchanges the two coordinates. Looks like reflection throughthe line x1 = x2:

(s, s)x

1

2

x

reflect

reflect

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 3, reflection through a line:Use

A =

[0 11 0

]T (

[x1x2

]) = A

[x1x2

]=

[0 11 0

] [x1x2

]=

[0x1 + 1 · x2

1 · x1 + 0 · x2

]=

[x2x1

]So T exchanges the two coordinates. Looks like reflection throughthe line x1 = x2:

(s, s)x

1

2

x

reflect

reflect

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 4, rotation:Use

A =

[cos θ − sin θsin θ cos θ

]

T (

[10

]) = A

[10

]=

[cos θ − sin θsin θ cos θ

] [10

]=

[cos θsin θ

]So horizontal unit vector is rotated c-clockwise an angle θ.

Similarly, for the vertical unit vector

[01

], so all of plane rotates:

v

R (v)

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 4, rotation:Use

A =

[cos θ − sin θsin θ cos θ

]T (

[10

]) = A

[10

]=

[cos θ − sin θsin θ cos θ

] [10

]=

[cos θsin θ

]

So horizontal unit vector is rotated c-clockwise an angle θ.

Similarly, for the vertical unit vector

[01

], so all of plane rotates:

v

R (v)

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 4, rotation:Use

A =

[cos θ − sin θsin θ cos θ

]T (

[10

]) = A

[10

]=

[cos θ − sin θsin θ cos θ

] [10

]=

[cos θsin θ

]So horizontal unit vector is rotated c-clockwise an angle θ.

Similarly, for the vertical unit vector

[01

], so all of plane rotates:

v

R (v)

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 4, rotation:Use

A =

[cos θ − sin θsin θ cos θ

]T (

[10

]) = A

[10

]=

[cos θ − sin θsin θ cos θ

] [10

]=

[cos θsin θ

]So horizontal unit vector is rotated c-clockwise an angle θ.

Similarly, for the vertical unit vector

[01

],

so all of plane rotates:

v

R (v)

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Example 4, rotation:Use

A =

[cos θ − sin θsin θ cos θ

]T (

[10

]) = A

[10

]=

[cos θ − sin θsin θ cos θ

] [10

]=

[cos θsin θ

]So horizontal unit vector is rotated c-clockwise an angle θ.

Similarly, for the vertical unit vector

[01

], so all of plane rotates:

v

R (v)

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Some types of problems that can come up:

Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?

Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.

Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that

T (~u) = ~b.

Answer: Solve the system of equations given by A~x = ~b. Any

solution is such a vector ~u.Reminder: There may be

no solution orexactly one solution ora parameterized family of solutions

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Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?

Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.

Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that

T (~u) = ~b.

Answer: Solve the system of equations given by A~x = ~b. Any

solution is such a vector ~u.Reminder: There may be

no solution orexactly one solution ora parameterized family of solutions

12/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?

Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.

Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that

T (~u) = ~b.

Answer: Solve the system of equations given by A~x = ~b. Any

solution is such a vector ~u.Reminder: There may be

no solution orexactly one solution ora parameterized family of solutions

12/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?

Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.

Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.

Find a vector ~u so thatT (~u) = ~b.

Answer: Solve the system of equations given by A~x = ~b. Any

solution is such a vector ~u.Reminder: There may be

no solution orexactly one solution ora parameterized family of solutions

12/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?

Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.

Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that

T (~u) = ~b.

Answer: Solve the system of equations given by A~x = ~b. Any

solution is such a vector ~u.Reminder: There may be

no solution orexactly one solution ora parameterized family of solutions

12/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?

Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.

Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that

T (~u) = ~b.

Answer: Solve the system of equations given by A~x = ~b. Any

solution is such a vector ~u.

Reminder: There may be

no solution orexactly one solution ora parameterized family of solutions

12/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?

Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.

Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that

T (~u) = ~b.

Answer: Solve the system of equations given by A~x = ~b. Any

solution is such a vector ~u.Reminder: There may be

no solution orexactly one solution ora parameterized family of solutions

12/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

iClicker question

Suppose T is a matrix linear transformation with matrix A below,and we are seeking all vectors ~u so that T (~u) = ~b.

A =

1 2 30 4 50 0 0

~b =

678

How many solutions are there?

A) Zero.

B) One

C) Infinity

D) Can’t tell

What if ~b =

670

?

13/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

iClicker question

Suppose T is a matrix linear transformation with matrix A below,and we are seeking all vectors ~u so that T (~u) = ~b.

A =

1 2 30 4 50 0 0

~b =

678

How many solutions are there?

A) Zero.

B) One

C) Infinity

D) Can’t tell

What if ~b =

670

?

13/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Question: Describe all vectors ~u so that T (~u) = ~b.

Answer: This is the same as finding all vectors ~u so that A~u = ~b.Could be no ~u, could be exactly one ~u, or could be a parametrizedfamily of such ~u’s.

Recall the idea: row reduce the augmented matrix [A : ~b] to merelyechelon form.

Augmentation column is pivot column ⇐⇒ no solutions.

Augmentation column is only non-pivot column ⇐⇒ uniquesolution.

There are free variables ⇐⇒ there is a parameterized familyof solutions whose dimension is the number of free variables.

If there are solutions, reduced echelon form makes it easy todescribe them.

14/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Question: Describe all vectors ~u so that T (~u) = ~b.

Answer: This is the same as finding all vectors ~u so that A~u = ~b.Could be no ~u, could be exactly one ~u, or could be a parametrizedfamily of such ~u’s.

Recall the idea: row reduce the augmented matrix [A : ~b] to merelyechelon form.

Augmentation column is pivot column ⇐⇒ no solutions.

Augmentation column is only non-pivot column ⇐⇒ uniquesolution.

There are free variables ⇐⇒ there is a parameterized familyof solutions whose dimension is the number of free variables.

If there are solutions, reduced echelon form makes it easy todescribe them.

14/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Question: Describe all vectors ~u so that T (~u) = ~b.

Answer: This is the same as finding all vectors ~u so that A~u = ~b.Could be no ~u, could be exactly one ~u, or could be a parametrizedfamily of such ~u’s.

Recall the idea: row reduce the augmented matrix [A : ~b] to merelyechelon form.

Augmentation column is pivot column ⇐⇒ no solutions.

Augmentation column is only non-pivot column ⇐⇒ uniquesolution.

There are free variables ⇐⇒ there is a parameterized familyof solutions whose dimension is the number of free variables.

If there are solutions, reduced echelon form makes it easy todescribe them.

14/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Question: Describe all vectors ~u so that T (~u) = ~b.

Answer: This is the same as finding all vectors ~u so that A~u = ~b.Could be no ~u, could be exactly one ~u, or could be a parametrizedfamily of such ~u’s.

Recall the idea: row reduce the augmented matrix [A : ~b] to merelyechelon form.

Augmentation column is pivot column ⇐⇒ no solutions.

Augmentation column is only non-pivot column ⇐⇒ uniquesolution.

There are free variables ⇐⇒ there is a parameterized familyof solutions whose dimension is the number of free variables.

If there are solutions, reduced echelon form makes it easy todescribe them.

14/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Question: Describe all vectors ~u so that T (~u) = ~b.

Answer: This is the same as finding all vectors ~u so that A~u = ~b.Could be no ~u, could be exactly one ~u, or could be a parametrizedfamily of such ~u’s.

Recall the idea: row reduce the augmented matrix [A : ~b] to merelyechelon form.

Augmentation column is pivot column ⇐⇒ no solutions.

Augmentation column is only non-pivot column ⇐⇒ uniquesolution.

There are free variables ⇐⇒ there is a parameterized familyof solutions whose dimension is the number of free variables.

If there are solutions, reduced echelon form makes it easy todescribe them.

14/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Definition

A linear transformation is a function T : Rn → Rm with theseproperties:

For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).

For any linear transformation T (~0) = ~0T (a~u + b~v) = aT (~u) + bT (~v)

This has important implications: if you know T (~u) and T (~v) ,then you know the values of T on all the linear combinations of ~uand ~v .

Matrix transformation: Let A be any m × n matrix. DefineT : Rn → Rm by T (~x) = A~x .

15/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Definition

A linear transformation is a function T : Rn → Rm with theseproperties:

For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).

For any linear transformation T (~0) = ~0

T (a~u + b~v) = aT (~u) + bT (~v)

This has important implications: if you know T (~u) and T (~v) ,then you know the values of T on all the linear combinations of ~uand ~v .

Matrix transformation: Let A be any m × n matrix. DefineT : Rn → Rm by T (~x) = A~x .

15/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Definition

A linear transformation is a function T : Rn → Rm with theseproperties:

For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).

For any linear transformation T (~0) = ~0T (a~u + b~v) = aT (~u) + bT (~v)

This has important implications: if you know T (~u) and T (~v) ,then you know the values of T on all the linear combinations of ~uand ~v .

Matrix transformation: Let A be any m × n matrix. DefineT : Rn → Rm by T (~x) = A~x .

15/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Definition

A linear transformation is a function T : Rn → Rm with theseproperties:

For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).

For any linear transformation T (~0) = ~0T (a~u + b~v) = aT (~u) + bT (~v)

This has important implications: if you know T (~u) and T (~v) ,then you know the values of T on all the linear combinations of ~uand ~v .

Matrix transformation: Let A be any m × n matrix. DefineT : Rn → Rm by T (~x) = A~x .

15/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Definition

A linear transformation is a function T : Rn → Rm with theseproperties:

For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)

For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).

For any linear transformation T (~0) = ~0T (a~u + b~v) = aT (~u) + bT (~v)

This has important implications: if you know T (~u) and T (~v) ,then you know the values of T on all the linear combinations of ~uand ~v .

Matrix transformation: Let A be any m × n matrix. DefineT : Rn → Rm by T (~x) = A~x .

15/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Matrix Is Everywhere

Example: Suppose T : R2 → R2 is a linear transformation so that

T (

[10

]) =

[52

]; T (

[01

]) =

[34

]

What is T (

[−17

])?

T (

[−17

]) = T (−1

[10

]+ 7

[01

]) = −1T (

[10

]) + 7T (

[01

])

= −1

[52

]+ 7

[34

]=

[1626

]In fact, nothing can stop us from using the same idea to compute

T (

[2−4

]) or T (~x) for any vector ~x ∈ R2:

Example

16/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Matrix Is Everywhere

Example: Suppose T : R2 → R2 is a linear transformation so that

T (

[10

]) =

[52

]; T (

[01

]) =

[34

]

What is T (

[−17

])?

T (

[−17

]) = T (−1

[10

]+ 7

[01

]) = −1T (

[10

]) + 7T (

[01

])

= −1

[52

]+ 7

[34

]=

[1626

]In fact, nothing can stop us from using the same idea to compute

T (

[2−4

]) or T (~x) for any vector ~x ∈ R2:

Example

16/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Matrix Is Everywhere

Example: Suppose T : R2 → R2 is a linear transformation so that

T (

[10

]) =

[52

]; T (

[01

]) =

[34

]

What is T (

[−17

])?

T (

[−17

]) = T (−1

[10

]+ 7

[01

]) = −1T (

[10

]) + 7T (

[01

])

= −1

[52

]+ 7

[34

]=

[1626

]

In fact, nothing can stop us from using the same idea to compute

T (

[2−4

]) or T (~x) for any vector ~x ∈ R2:

Example

16/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Matrix Is Everywhere

Example: Suppose T : R2 → R2 is a linear transformation so that

T (

[10

]) =

[52

]; T (

[01

]) =

[34

]

What is T (

[−17

])?

T (

[−17

]) = T (−1

[10

]+ 7

[01

]) = −1T (

[10

]) + 7T (

[01

])

= −1

[52

]+ 7

[34

]=

[1626

]In fact, nothing can stop us from using the same idea to compute

T (

[2−4

]) or T (~x) for any vector ~x ∈ R2:

Example16/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

We can carry this much further: All linear transformationsT : Rn → Rm are matrix linear transformations

Why?

~x =

x1x2...xn

= x1

10...0

+ x2

01...0

+ . . . xn

00...1

is a linear combination of the vectors

10...0

,

01...0

, . . . ,

00...1

(standard basis of Rn)

17/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

We can carry this much further: All linear transformationsT : Rn → Rm are matrix linear transformations

Why?

~x =

x1x2...xn

= x1

10...0

+ x2

01...0

+ . . . xn

00...1

is a linear combination of the vectors

10...0

,

01...0

, . . . ,

00...1

(standard basis of Rn)

17/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

So by the property of linear transformation

T (~x) = x1T

10...0

+ x2T

01...0

+ . . . xnT

00...1

only need to know each T (~ej) where

~ej =

0...1...0

← j thentry

Denote ~aj = T (~ej)

T (~x) = x1 ~a1 + x2 ~a2 + . . . xn ~an

18/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

So by the property of linear transformation

T (~x) = x1T

10...0

+ x2T

01...0

+ . . . xnT

00...1

only need to know each T (~ej) where

~ej =

0...1...0

← j thentry Denote ~aj = T (~ej)

T (~x) = x1 ~a1 + x2 ~a2 + . . . xn ~an

18/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Ahha: In matrix notation this is written:

T (~x) =[~a1 ~a2 . . . ~an

]x1x2...xn

= A~x

That is, the matrix

A =[~a1 ~a2 . . . ~an

]is the matrix of T !

19/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Ahha: In matrix notation this is written:

T (~x) =[~a1 ~a2 . . . ~an

]x1x2...xn

= A~x

That is, the matrix

A =[~a1 ~a2 . . . ~an

]is the matrix of T !

19/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Recap:

Since T is a linear transformation,

T (~x) = T (x1~e1+x2~e2+. . . xn~en) = x1T (~e1)+x2T (~e2)+. . . xnT (~en) =

x1~a1 + x2~a2 + . . . xn~an =[~a1 ~a2 . . . ~an

]

x1x2...xn

= A~x .

Example

20/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Recap: Since T is a linear transformation,

T (~x) = T (x1~e1+x2~e2+. . . xn~en) = x1T (~e1)+x2T (~e2)+. . . xnT (~en) =

x1~a1 + x2~a2 + . . . xn~an =[~a1 ~a2 . . . ~an

]

x1x2...xn

= A~x .

Example

20/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Recap: Since T is a linear transformation,

T (~x) = T (x1~e1+x2~e2+. . . xn~en) = x1T (~e1)+x2T (~e2)+. . . xnT (~en) =

x1~a1 + x2~a2 + . . . xn~an =[~a1 ~a2 . . . ~an

]

x1x2...xn

= A~x .

Example

20/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Recap: Since T is a linear transformation,

T (~x) = T (x1~e1+x2~e2+. . . xn~en) = x1T (~e1)+x2T (~e2)+. . . xnT (~en) =

x1~a1 + x2~a2 + . . . xn~an =[~a1 ~a2 . . . ~an

]

x1x2...xn

= A~x .

Example

20/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).

Put another words, T (~u) = T (~v) implies ~u = ~v .

If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).

Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections

21/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).

Put another words, T (~u) = T (~v) implies ~u = ~v .

If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).

Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections

21/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).

Put another words, T (~u) = T (~v) implies ~u = ~v .

If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).

Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections

21/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).

Put another words, T (~u) = T (~v) implies ~u = ~v .

If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).

Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections

21/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).

Put another words, T (~u) = T (~v) implies ~u = ~v .

If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).

Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.

The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections

21/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).

Put another words, T (~u) = T (~v) implies ~u = ~v .

If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).

Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.

Examples: shears, contractions and expansions, rotations,reflections

21/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).

Put another words, T (~u) = T (~v) implies ~u = ~v .

If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).

Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections

21/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is onto if, for any ~v ∈ Rm, is there a ~u ∈ Rnsuch thatT (~u) = ~v .

Saying that T is onto is the same as saying thatT (~u) = ~v always has a solution.

This means that the reduced echelon form of the matrix of T musthave exactly m non-zero rows.The non-homogeneous equation must always have a solution. And,as a consequence, m ≤ n.Examples: shears, contractions and expansions, rotations,reflections

22/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is onto if, for any ~v ∈ Rm, is there a ~u ∈ Rnsuch thatT (~u) = ~v .

Saying that T is onto is the same as saying thatT (~u) = ~v always has a solution.

This means that the reduced echelon form of the matrix of T musthave exactly m non-zero rows.The non-homogeneous equation must always have a solution. And,as a consequence, m ≤ n.Examples: shears, contractions and expansions, rotations,reflections

22/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is onto if, for any ~v ∈ Rm, is there a ~u ∈ Rnsuch thatT (~u) = ~v .

Saying that T is onto is the same as saying thatT (~u) = ~v always has a solution.

This means that the reduced echelon form of the matrix of T musthave exactly m non-zero rows.

The non-homogeneous equation must always have a solution. And,as a consequence, m ≤ n.Examples: shears, contractions and expansions, rotations,reflections

22/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is onto if, for any ~v ∈ Rm, is there a ~u ∈ Rnsuch thatT (~u) = ~v .

Saying that T is onto is the same as saying thatT (~u) = ~v always has a solution.

This means that the reduced echelon form of the matrix of T musthave exactly m non-zero rows.The non-homogeneous equation must always have a solution. And,as a consequence, m ≤ n.

Examples: shears, contractions and expansions, rotations,reflections

22/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rm is a (matrix) linear transformation.

Definition

T is onto if, for any ~v ∈ Rm, is there a ~u ∈ Rnsuch thatT (~u) = ~v .

Saying that T is onto is the same as saying thatT (~u) = ~v always has a solution.

This means that the reduced echelon form of the matrix of T musthave exactly m non-zero rows.The non-homogeneous equation must always have a solution. And,as a consequence, m ≤ n.Examples: shears, contractions and expansions, rotations,reflections

22/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rn is a (matrix) linear transformation.

Definition

T is an isomorphism if T is both 1 to 1 and onto.

Saying that T is isomorphism is the same as saying thatT is a bijection that respects the vector space structure.

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows, the same as the number of columns.

The non-homogeneous equation must always have exactly onesolution.

Examples: shears, contractions and expansions, rotations,reflections

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rn is a (matrix) linear transformation.

Definition

T is an isomorphism if T is both 1 to 1 and onto.

Saying that T is isomorphism is the same as saying thatT is a bijection that respects the vector space structure.

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows, the same as the number of columns.

The non-homogeneous equation must always have exactly onesolution.

Examples: shears, contractions and expansions, rotations,reflections

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rn is a (matrix) linear transformation.

Definition

T is an isomorphism if T is both 1 to 1 and onto.

Saying that T is isomorphism is the same as saying thatT is a bijection that respects the vector space structure.

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows, the same as the number of columns.

The non-homogeneous equation must always have exactly onesolution.

Examples: shears, contractions and expansions, rotations,reflections

23/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rn is a (matrix) linear transformation.

Definition

T is an isomorphism if T is both 1 to 1 and onto.

Saying that T is isomorphism is the same as saying thatT is a bijection that respects the vector space structure.

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows, the same as the number of columns.

The non-homogeneous equation must always have exactly onesolution.

Examples: shears, contractions and expansions, rotations,reflections

23/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Suppose T : Rn → Rn is a (matrix) linear transformation.

Definition

T is an isomorphism if T is both 1 to 1 and onto.

Saying that T is isomorphism is the same as saying thatT is a bijection that respects the vector space structure.

This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows, the same as the number of columns.

The non-homogeneous equation must always have exactly onesolution.

Examples: shears, contractions and expansions, rotations,reflections

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.

T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.

T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.

T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.

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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.

T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.

Said differently, the column vectors of the matrix of T are linearlyindependent.

T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.

T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.

24/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.

T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.

T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.

T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.

24/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.

T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.

T is onto if there is a pivot 1 in every row of the reduced echelonform.

Said differently, the column vectors of the matrix of T span thewhole space Rm.

T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.

24/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.

T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.

T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.

T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.

24/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.

T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.

T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.

T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.

Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.

24/24

Prelude Linear Transformations Pictorial examples Matrix Is Everywhere

Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.

T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.

T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.

T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.

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