1. 4.3 Matrix of Linear Transformations y RS T R ′S ′ T ′ x

2. Example 2 Find a matrix B that represents a linear transformation from T(f) = f ’ + f ” from P 2 to P 2 with respect to the standard basis Β =(1, x, x 2 )

3. Example 2 T(f) = f ’ + f ”

4. Similar matrices Had we used a different basis, we could describe this same transformation using that basis. Two matrices that describe the same transformation with regard to a different basis are called similar matrices and are related by the formula SAS -1 = B In this formula A is similar to B

5. An Application Write a matrix that will find the 2 nd derivative of a polynomial of degree 3 or lower. Use this matrix to find the 2 nd derivative of x 3 + 2x 2 + 4x +1

6. Application solution Start with a basis: 1,x,x 2,x 3 Find the second derivative of each of the elements of the basis. Write the answer in terms of coordinates of the basis. 0 0 2 0 0 0 0 6 0 0

7. Application part B Use matrix multiplication to find the second derivative of x 3 + 2x 2 + 4x +1

8. Application part B Multiply the coordinate matrix times the matrix that represents x 3 + 2x 2 + 4x +1 in terms of our basis 1,x,x 2,x 3 0 0 2 0 1 4 0 0 0 6 4 = 6 0 0 0 0 2 0 0 0 0 0 1 0 y ” =6x + 4

9. A Matrix of transformation

10. Forming a Matrix of transformation

11. Example 3

12. Solution to 3a

13. Solution to 3b Because there is an invertible matrix that describes the transformation T we call T an isomorphism

14. Problem 6 Find the matrix of transformation

15. 6 solution What does this mean? If I had the vector as my x it means that I had 1 of the first element, 0 of the second and 1 of the 3 rd Or the matrix 1 1 and ran it through the transformation I would get 0 1 The matrix 1 3 using the answer from above as A as x 0 3 yields which are coordinates for the answer [ ]

16. Homework p.181 1-19 odd lim sin(x) = 6 n --> ∞ n Proof: cancel the n in the numerator and denominator.

17. Example 1 Express using coordinates

18. Example 1 Solution

20. Example 4

21. Solution to example 4 a

22. Solution to example 4b

23. Solution to 4 c

24. Example 6

25. Example 6 solution

26. Why are similar matrices related by B = S -1 AS Note: start at the lower left hand corner of the diagram and move to the upper right hand corner by each direction