Linear Transformations

Experiment with the transformation xA x where A = (0 -1) 1 1.

Move the vector x in the plot on the left .        You'll see A x move in the plot on the right.

Check that A (2) = (-1) 1 3 by moving x to (2) 1.   Is A x on (-1) 3?

(Q1) Now compute A (-2) 1 by moving x to (-2) 1.  Answer: A (-2) = 1

Check that A sends (1) 3to (-3) 4by moving x to (1) 3.  Is A x on (-3) 4?

(Q2) Where does A send the vector (3) 1?

Suppose that for the matrix A = (0 -1) 1 1you want to solve A x = (1) 1.

The old way of thinking about this problem would be to row reduce (0 -1 1 ) 1 1 1.  
The new way of thinking about this problem asks you to find the x that A sends to (1) 1.

Here's the same picture of xA x with y = (1) 1 added.

Move the vector x until you find the x that A sends to (1) 1.

If you found the answer x = (2 ) -1, then you've opened your mind up to a whole new way of solving A x = y .

Here's the same picture of xA x with y = (-3) 2 added.

(Q3) Use the graphic to find x that solves A x = (-3) 2.

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Copyright © 2007 Todd Will
Created by Mathematica  (April 15, 2007)