linear-guest

228 Eigenvalues and Eigenvectors
It was picked at random by choosing a pair of vectors L(e 1 ) and L(e 2 ) as
the outputs of L acting on the canonical basis vectors. Notice how the unit
square with a corner at the origin is mapped to a parallelogram. The second
line of the picture shows these superimposed on one another. Now look at the
second picture on that line. There, two vectors f 1 and f 2 have been carefully
chosen such that if the inputs into L are in the parallelogram spanned by f 1
and f 2 , the outputs also form a parallelogram with edges lying along the same
two directions. Clearly this is a very special situation that should correspond
to interesting properties of L.
Now lets try an explicit example to see if we can achieve the last picture:
Example 126 Consider the linear transformation L such that
( ( ) ( ( 1 −4 0 3
L = and L = ,
0)
−10 1)
7)
so that the matrix of L in the standard basis is
( ) −4 3
.
−10 7
( ( 1 0
Recall that a vector is a direction and a magnitude; L applied to or changes
0)
1)
both the direction and the magnitude of the vectors given to it.
Notice that ( ( ) ( 3 −4 · 3 + 3 · 5 3
L =
= .
5)
−10 · 3 + 7 · 5 5)
228