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)