3.1 Day 2 Applications and properties of a Kernel

3.1 day 2 Applications and properties of a Kernel

…Wait…

No, wrong slide…

not that kind of a kernel…

Properties of invertible matrices

The following statements are equivalent for an nxn matrix

1 A is invertible2 The linear system Ax = b has a unique solution x

for all b3 rref(A) = I4 Im(A) = Rn

5 Rank (A) = n6 Ker(A) = 0

Properties of an image of a Linear Transformation

1. The zero vector is in the image of T

2. The image of T is closed under addition.

(if two vectors are in the image of T then the sum of those two vectors is also in the image of T)

3. The image of T is closed under scalar multiplication. if a vector is in the image then any scalar multiple of that vector is also in the image)

Prove Ker(A)= 0 for all invertible matrices

Problem 22

Find the image by both inspection and rref

Solution to Problem 22

To see if the last column is a scalar multiple of the first two use2x +y =3 x =2 y = -13x+4y = 2Find the values of x and y and see if they work in the equation6x+5y = 7

Example 7

Example: 7 Solution

Use your library of basic matrices to find a basis for the kernel and image

Find the kernel and image of a matrix in R2

9/25 12/25

12/25 16/25

Use your library of basic matrices to find a basis for the kernel and image

Find the kernel and image of a matrix in R2

9/25 12/25

12/25 16/25

(Hint this is a projection matrix onto <3,4>

Use your library of basic matrices to find a basis for the kernel and image

Find the kernel and image of a matrix in R2

9/25 12/25

12/25 16/25

This matrix is a projection matrix onto the line that consists of the scalar multiples of <3,4>

The image is the line (the scalar multiples of <3,4>)The kernel are vectors that are perpendicular to the given line.

Find the image and kernel of the following matrix

-3/5

3/5 4/5

4/5

Use your library of basic matrices to obtain the answer

Find the image and kernel of the following matrix

-3/5

3/5 4/5

4/5This is a rotation matrix the image is R2

The Kernel is the zero vector

Problem 24

Describe the Image and the Kernel of the transformation

Problem 24 Solution

Problem 37

37 Solution

Problem 38

38 Solution

Homework p. 110 23,25,27,29,39,41,44,49

1. Snakes on a plane2. Independence day3. Mean Girls4. Paths of glory5. 8 Mile6. Matrix Revolutions7. Field of dreams8. Alpha & Omega9. 13th Floor10. Sum of all fears