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Linear Algebra Lecture 10

Linear Algebra Lecture 10. Systems of Linear Equations. The Matrix of a Linear Transformations. Every Linear Transformation from R n to R m is actually a matrix Transformation. Example 1. Theorem. Let

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Linear Algebra Lecture 10

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  1. Linear Algebra Lecture 10

  2. Systems of Linear Equations

  3. The Matrix of a Linear Transformations

  4. Every Linear Transformation from Rn to Rm is actually a matrix Transformation

  5. Example 1

  6. Theorem Let be a linear transformation. Then there exist a unique matrix A such that T(x)=Ax for all x in Rn.

  7. Example 2 Find the standard matrix A for the dilation transformation T(x)=3x for all x in R2

  8. Example 3 Let be the linear operator defined by Find the standard matrix representing L and verify L(x)=Ax

  9. Example 4

  10. Geometric LT of R2 Examples 2 and 3 illustrate LT that are described geometrically. Tables 1 – 4 in the book illustrate other common geometric linear transformations of the plane. Because the transformations are linear, they are determined completely by what they do to the columns of I2

  11. Instead of showing only the images of e1and e2, the tables show what a transformation does to the unit square

  12. Other transformations can be constructed from those listed in Tables 1 – 4 by applying one transformation after another

  13. Example 5

  14. Definitions

  15. Example 6

  16. Theorems

  17. Example 7

  18. Linear Algebra Lecture 10

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