1. 3.1 Day 2 Applications and properties of a Kernel

2. 3.1 day 2 Applications and properties of a Kernel …Wait… No, wrong slide… not that kind of a kernel…

3. Properties of invertible matrices The following statements are equivalent for an nxn matrix 1 A is invertible 2 The linear system Ax = b has a unique solution x for all b 3rref(A) = I 4Im(A) = R n 5Rank (A) = n 6Ker(A) = 0

4. 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)

5. Prove Ker(A)= 0 for all invertible matrices

6. Problem 22 Find the image by both inspection and rref

7. Solution to Problem 22 To see if the last column is a scalar multiple of the first two use 2x +y =3 x =2 y = -1 3x+4y = 2 Find the values of x and y and see if they work in the equation 6x+5y = 7

8. Example 7

9. Example: 7 Solution

10. Use your library of basic matrices to find a basis for the kernel and image Find the kernel and image of a matrix in R 2 9/25 12/25 12/25 16/25

11. Use your library of basic matrices to find a basis for the kernel and image Find the kernel and image of a matrix in R 2 9/25 12/25 12/25 16/25 (Hint this is a projection matrix onto

12. Use your library of basic matrices to find a basis for the kernel and image Find the kernel and image of a matrix in R 2 9/25 12/25 12/25 16/25 This matrix is a projection matrix onto the line that consists of the scalar multiples of The image is the line (the scalar multiples of ) The kernel are vectors that are perpendicular to the given line.

13. 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

14. Find the image and kernel of the following matrix -3/5 3/5 4/5 4/5 This is a rotation matrix the image is R 2 The Kernel is the zero vector

15. Problem 24 Describe the Image and the Kernel of the transformation

16. Problem 24 Solution

17. Problem 37

18. 37 Solution

19. Problem 38

20. 38 Solution

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

22. 1. Snakes on a plane 2. Independence day 3. Mean Girls 4. Paths of glory 5. 8 Mile 6. Matrix Revolutions 7. Field of dreams 8. Alpha & Omega 9. 13th Floor 10. Sum of all fears