2.1 Day 2 Linear Transformations and their Inverses For an animation of this topic visit: Multiply the coordinates of the light blue dog by A to get the dark blue dog A= Note: view each coordinate separately as the equation Ax=b If the coordinates (of the dog) are put together to make a matrix, the equation takes Form AB= C

Example To encode the position of the dog on the previous slide multiply each coordinate of the dog by the matrix in the form Ax = b (A is the matrix) x is a vertical vector with components of the coordinates of the dog. b is the new coordinate. This is called a transformation matrix

Example If we want to start with the dark blue dog and end with the light blue dog we will need to multiply the coordinates of the dark blue dog by a matrix. This matrix is called the inverse of Matrix A. We learned how to compute an inverse yesterday.

Find the inverse What if we had multiplied the coordinates of the dog by the following matrix What would the inverse be? What would the transformation look like? (answer on next slide)

The graph What would the inverse of this matrix be? Why?

Problem 13 a Prove that for the given matrix, it is invertible if and only if the determinant is not = 0 Hint prove the case of a = 0 and separately

Problem 13 solution

Problem 13 solution continued

Linear Transformations This represents a transformation from R 3 to R

Linear transformations If there is a n nxm matrix A such that T(x) = A(x) For all x in

Problem 2 Is this transformation linear? What is the matrix of transformation? How could this transformation be changed to make it non-linear?

Problem 2 Solution Possible solution to make system non-linear y 1 = x 1 y 2 (is non-linear due to multiplication of variables) y 2 = x 2 +1 (is non-linear due to adding a constant (note: this is different from other courses you have taken) y 3 = (x 1 ) 2 (non-linear due to an exponent) If any one of these is non-linear than the entire expression is non-linear

Problem 16 Describe the transformation. And the inverse of the transformation Note: when you do your homework keep a record of each type of transformation

Problem 16 Solution

Rectangular matrices A 3 x 3 matrix represents a transformation from R 3 to R 3 What kind of transformation does a 2 x 3 matrix represent? What kind of transformation does an mxn matrix represent?

Rectangular matrices A 3 x 3 matrix represents a transformation from R 3 to R 3 What kind of transformation does a 2 x 3 matrix represent? R 3 to R 2 What kind of transformation does an mxn matrix represent? R n to R m

Homework p all (13 was already done as part of the lesson) all (May use the website shown on first slide to help with the geometric interpretation) Some math quotes: The Romans didn't find algebra very challenging, because X was always 10. (Aaron Dragushan) “There are three kinds of lies: lies, damned lies, and statistics.” Attributed by Mark Twain to Benjamin Disraeli.