Page 392 - 35Linear Algebra

392 Movie Scripts

3 2
Z 2 or Z 2

Now lets consider a linear transformation
3 2
L : Z −→ Z .
2 2
This must be represented by a matrix, and lets take the example

   
x x
0 1 1
L  y  =  y  := AX .
1 1 0
z z
Since we have bits, we can work out what L does to every vector, this is listed
below
L
(0, 0, 0) 7→ (0, 0)
L
(0, 0, 1) 7→ (1, 0)
L
(1, 1, 0) 7→ (1, 0)
L
(1, 0, 0) 7→ (0, 1)
L
(0, 1, 1) 7→ (0, 1)
L
(0, 1, 0) 7→ (1, 1)
L
(1, 0, 1) 7→ (1, 1)
L
(1, 1, 1) 7→ (1, 1)
Now lets think about left and right inverses. A left inverse B to the matrix
A would obey
BA = I

and since the identity matrix is square, B must be 2 × 3. It would have to
3
undo the action of A and return vectors in Z to where they started from. But
2
3 2
above, we see that different vectors in Z are mapped to the same vector in Z
2 2
by the linear transformation L with matrix A. So B cannot exist. However a
right inverse C obeying
AC = I
2 3
can. It would be 2×2. Its job is to take a vector in Z back to one in Z in a
2
2
way that gets undone by the action of A. This can be done, but not uniquely.
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