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Inverse of Linear Transformations

Learn about invertible functions, matrix invertibility, finding inverses, and applications of cryptography using inverses in this comprehensive guide.

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Inverse of Linear Transformations

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  1. 2.4 Inverse of Linear Transformations Is the transformation depicted In this picture invertible? For an animation of this topic visit: http://www.maa.org/joma/Volume7/Hohenwarter/Transformations.html

  2. Ax=b Note our standard equation for the course is Ax=b However today we will look at the form xA=b In this case x must be a row vector to multiply by a matrix

  3. Invertible Functions A function T from X to Y is called invertible if the equation T(x) = y has a unique solution x in X for each y in Y

  4. Note: Inverses are only defined for Square matrices Invertibility An nxn matrix is invertible if and only if • rref of A is I • Det(A) ≠ 0 • There are no vectors other than the zero vector that satisfy the equation Ax=0 • No row of A is a multiple of another row. No column of A is a multiple of another column No row of A is a linear combination of other rows of A No column of A is a linear combination of other columns of A e) rank (A) = n A non invertible matrix is called a singular matrix

  5. How to find an inverse In previous classes, we showed how to find the inverse of a matrix. We will not review that here. However, at there are 2 examples with step by step instructions of how to find the inverse of a matrix that you can review if you choose.

  6. CryptographyAn application of Inverses

  7. Cryptography A Cryptogram is a message written according to a secret code. (The Greek word kryptos means hidden) If one wanted to write a secret message, one might first start by assigning a number to each letter of the alphabet (as shown on the next slide)

  8. Encryption One might then break up a message into groups of letters for this example we will use blocks of 3 Next multiply each sequence by an encryption matrix

  9. Continue this for each group of 3 terms

  10. How would one decode this message?

  11. One could use inverses to get the original terms back What would cause this system to not work properly? Please note that we are multiplying A-1 on the right side

  12. Problem 26Find the InverseNote: you must follow a different process than the one taught previously,Why does our previous method fail to work?

  13. 26 Solution

  14. Recall:

  15. How does the determinant of A relate to the determinant of A-1 Det(A) = 1/(detA-1)

  16. Homework P. 88 1-35 odd, 40, Pre-Calc book P. 608 29 - 39 odd,

  17. Example 1Find the inverse if it exists

  18. Example 1 Solution Inverse does not exist

  19. Example 2Find the inverse if it exists

  20. Example 2 Solution

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