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4.3 Matrix of Linear Transformations

y. T. R. S. x. ′. ′. ′. R. S. T. 4.3 Matrix of Linear Transformations. 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 ). Example 2. T(f) = f ’ + f ”. Similar matrices.

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4.3 Matrix of Linear Transformations

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

  2. Example 2 Find a matrix B that represents a linear transformation from T(f) = f’ + f” from P2 to P2 with respect to the standard basis Β =(1, x, x2)

  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 2nd derivative of a polynomial of degree 3 or lower. Use this matrix to find the 2nd derivative of x3 + 2x2 + 4x +1

  6. Application solution Start with a basis: 1,x,x2,x3 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 0 0 0 0 0 0

  7. Application part B Use matrix multiplication to find the second derivative of x3 + 2x2 + 4x +1

  8. Application part B Multiply the coordinate matrix times the matrix that represents x3 + 2x2 + 4x +1 in terms of our basis 1,x,x2,x3 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 <1,0,1> as my x it means that I had 1 of the first element, 0 of the second and 1 of the 3rd 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 <1,01> as x 0 3 yields <1,0,3> 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

  19. 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-1AS Note: start at the lower left hand corner of the diagram and move to the upper right hand corner by each direction

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