math111_logo 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 ] : R3R3
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.


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[Extra: Root-Polynomial transformation]