math111_logoLinear Transformation

5. Linear transformation = matrix transformation

We have seen matrix transformations are linear. Conversely, this example suggests the converse is also true.

Any linear transformation T: RnRm is also a matrix transformation, with the matrix given by

A = [T(e1) T(e2) ... T(en)].

In other words, the columns of the matrix A are the values of the linear transformation T at the standard basis. Thus the concepts of matrix transformations and linear transformations (between euclidean spaces) are equivalent.

Example The identity transformation id: RnRn is clearly linear. The corresponding matrix is

[id(e1) id(e2) ... id(en)] = [e1 e2 ... en] = [ 1 0 . . . 0 ]
0 1 . . . 0
: :   :
0 0 . . . 1

called the identity matrix and denoted by In.

Example In this example, we started from a matrix

A = [ 1 -1 1 ]
3 1 -1
2 2 1

and produced a matrix transformation (which is also a linear transformation).

T[ x1 ] = [ x1 - x2 + x3 ] : R3R3
x2 3x1 + x2 - x3
x3 2x1 + 2x2 + x3

Conversely, we may recover the columns of A as follows

[col 1] of A = T[ 1 ] = [ 1 - 0 + 0 ] = [ 1 ]
0 3 + 0 - 0 3
0 2 + 0 + 0 2
[col 2] of A = T[ 0 ] = [ 0 - 1 + 0 ] = [ -1 ]
1 0 + 1 - 0 1
0 0 + 2 + 0 2
[col 3] of A = T[ 0 ] = [ 0 - 0 + 1 ] = [ 1 ]
0 0 + 0 - 1 -1
1 0 + 0 + 1 1

Example The reflections and rotations on R2 are linear transformations because they preserve parallelogram and stretching/shrinking. To find their matrices, we simply need to apply reflections and rotations to the standard basis.

reflection of standard basis

The picture shows that the reflections of e1, e2 in x-axis are e1 = (1, 0), -e2 = (0, -1). Therefore the matrix for the reflection is

[ 1 0 ]
0 -1

Moreover, we also see from picture that for the rotation by angle θ, T(e1) = (cosθ, sinθ), T(e2) = (-sinθ, cosθ).

rotation of standard basis

Therefore the matrix for the rotation is

[ cosθ - sinθ ]
sinθ cosθ

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[Extra: Matrix of a linear transformation]