Linear Transformation

2. Matrix transformation

We may think of the left side of a system of linear equations as a transformation. For example, the left side of the system

 x1 + 2x2 + 3x2 = b1 4x1 + 5x2 + 6x4 = b2

can be considered as the linear transformation T in the example above. By expressing vectors in vertical form, we also see that

 T[ x1 ] x2 x3
=
 [ x1 + 2x2 + 3x3 ] 4x1 + 5x2 + 6x3
=
 [ 1 2 3 ] 4 5 6
 [ x1 ] x2 x3

In particular, the formula for the transformation T is given by a matrix A:

T(x) = Ax.

In general, for any m by n matrix A, we may construct a transformation T: RnRm by the above formula. Such transformations are called matrix transformations.

Example The following matrices

 [ 1 -1 1 ] 3 1 -1 2 2 1
 [ 1 3 2 0 1 ] -1 -1 -1 1 0 0 4 2 4 3 1 3 2 -2 0

give us matrix transformations,

 T1[ x1 ] = [ x1 - x2 + x3 ] : R3 → R3 x2 3x1 + x2 - x3 x3 2x1 + 2x2 + x3
 T1[ x1 ] x2 x3 x4 x5 x6
=
 [ x1 + 3x2 + 2x3 + x5 ] - x1 - x2 - x3 + x4 + x6 4x2 + 2x3 + 4x4 + 3x5 + 3x6 x1 + 3x2 + 2x3 - 2x4
: R6R4

or written horizontally,

T1: R3R3, (x1, x2, x3) → (x1 - x2 + x3, 3x1 + x2 - x3, 2x1 + 2x2 + x3)

T2: R6R4, (x1, x2, x3, x4, x5) → (x1 + 3x2 + 2x3 + x5, - x1 - x2 - x3 + x4 + x6, 4x2 + 2x3 + 4x4 + 3x5 + 3x6, x1 + 3x2 + 2x3 - 2x4)

Example The transformations

S: R3R3, (x1, x2, x3) → (x1 - x2 + x3, 3x1 + x2 - x3 - 2, 2x1 + 2x2 + x3)

T: R2R2, (x, y) → (xy, x2)

are not matrix transformations because they cannot be expressed as Ax for a constant matrix A.