Lecture 7Hyper-planes, Matrices, and

Linear SystemsShang-Hua Teng

Line in 2D

• By linear equation

12 yx

12 yx x=3y=1

Line in 2D

• By a point and a vector: passing (3,1) along vector (2,1)

ty

tx

ttyx

1

23

:)1,2()1,3(),(

set edParametriz

12

have we, geliminatinBy

yx

t

x=3y=1

Line in 2D

• By two points: passing (3,1) and (0,-1/2)

ttty

x

tty

x

:5.0

0

1

3)1(

have we term,same theCombine

:1

3

5.0

0

1

3

set edParametriz

12

have we, geliminatinBy

yx

t

(3,1)

(0,-1/2)

Line and Affine Combination in 2D

• The line passing two points or the affine combination of two points is given by

tb

at

b

at

b

a

b

a

b

a

b

a

:)1(

,affine,line

points twoofn Combinatio Affineor Line

2

2

1

1

2

2

1

1

2

2

1

1

System of Linear Equations (2D)

• Row Picture[conventional view]: two lines meets at a point

1123

12

yx

yx

1123 yx

12 yx x=3y=1

System of Linear Equations (2D)

• Column Picture: linear combination of the first two vectors produces the third vector

1123

12

yx

yx

11

1

2

2

3

1yx

And geometrically• Column Picture: linear combination of the

first two vector produce the third vector

11

1

2

2

3

1yx

2

2

11

1

3

1

3

13

x=3y=1

Coefficient Matrix and Matrix-Vector Product

1123

12

yx

yx

11

1

2

2

3

1yx

11

1

23

21

y

x

A 2 by 2 matrix is a square table of 4 numbers, two per row and two per column

System of Linear Equations (3D)

• Row Picture[conventional view]: Three planes meet at a single point

236

4252

632

zyx

zyx

zyx

2

0

0

• Row Picture[conventional view]: Two planes meet at a single line A line and a plane meet at a single point

Intersection of Planes

System of Linear Equations (3D)

• Column Picture: linear combination of the first three vectors produces the fourth vector

2

4

6

1

2

3

3

5

2

6

2

1

zyx

2

0

0

236

4252

632

zyx

zyx

zyx

Coefficient Matrix and Matrix-Vector Product

2

4

6

136

252

321

z

y

x

A 3 by 3 matrix is a square table of 9 numbers, three per row and three per column

236

4252

632

zyx

zyx

zyx

2

4

6

1

2

3

3

5

2

6

2

1

zyx

Matrix Vector Product (by row)

• If A is a 3 by 3 matrix and x is a 3 by 1 vector, then in the row picture

xrow

xrow

xrow

Ax

)2 (

)2 (

)1 (

Matrix Vector Product (by column)

• If A is a 3 by 3 matrix and x is a 3 by 1 vector, then in the row picture

3

2

1

321

where

)3 ()2 ()1 (

x

x

x

x

columnxcolumnxcolumnxAx

More about 3D Geometry

• Points and distance, Balls and Spheres– 0 dimension in 3 dimensions

• Lines– 1 dimension in 3 dimensions

• Plane– 2 dimensions in 3 dimensions

Line in 3D

• 2D– By linear equation – A point and a vector– Two points

• Affine combination

• 3D– A point and a vector– Two points

• Affine combination

Line in 3D

• By a point and a vector: passing p along vector v

zz

yy

xx

tvpz

tvpy

tvpx

ttvp

:

Line and Affine Combination in 3D• The line passing two points or the affine combination of

two points is given by

t

v

vv

t

u

uu

t

v

vv

u

uu

v

vv

u

uu

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

:)1(

,affine,line

points twoofn Combinatio Affineor Line

Plane in 3D

• Line in 2D– By linear equation – Affine combination of two points

• “Every” two points determine a line

• 3D– By linear equation– Affine combination of three points

• “Every” three points determine a plane

Linear Equation and its Normal

3

2

1

so

032

implying

632

632

have, we,, and ,, :on points any twofor

632:),,(

21

21

21

212121

222

111

222111

zz

yy

xx

zzyyxx

zyx

zyx

zyxzyxP

zyxzyxP

Normal of a Plane

Plane and Affine Combination in 3D

)1(

,,affine

,,plane

321

321

321

ppp

ppp

ppp

1p

2p

3p

u

v

s

tsst

psptsstppssuv

pttpu

)1(,

)1()1( )1(

)1(

3213

21

High Dimensional Geometric Extension• Points and distance, Balls and Spheres

– 0 dimension in n dimensions

• Lines– 1 dimension in n dimensions

• Plane– 2 dimensions in n dimensions

• k-flat – k-dimensions in n dimensions

• Hyper-plane – (n-1)-dimensions in n dimensions

Affine Combination in n-D

1

,,,affine

1

12211

21

k

k

j j

k

ppp

ppp

Hyper-Planes in d-D

• Line in 2D– By linear equation – Affine combination of two points

• 3D– By linear equation– Affine combination of three points

• n-D– By linear equation– Affine combination of n-1 points

Linear Equation and its Normal

n

n

k kkk

n

k kk

n

k kk

nn

n

k kkn

aaaa

where

ayx

yxa

bya

bxa

yyyxxxP

bxaxxxP

,,,

)(

so

0)(

implying

, ,,,y and ,,, x:on points Any two

:),,,(

21

1

1

1

2121

121

Matrix (Uniform Representation for Any Dimension)

• An m by n matrix is a rectangular table of mn numbers

ji

nmmm

n

n

ajiA

aaa

aaa

aaa

A

,

,2,1,

,22,21,2

,12,11,1

),( write weSometime

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