Definition 2.7.1.
Suppose that \(R\) is a subspace of \(\real^m\) and \(S\) is a subspace of \(\real^n\text{.}\) Then a linear transformation \(T : R \to S \) is a function that satisfies the two linearity properties from Assemblage .
Section 2.7 Coordinate representation of linear transformations
In this section, we will generalize our discussion of linear transformations to linear transformations between subspaces.
Subsection 2.7.1 Linear transformations between subspaces
Suppose that \(\bcal = \begin{bmatrix} \vvec_1 & \cdots
& \vvec_p \end{bmatrix}\) is a basis of \(R\) and \(\ccal =
\begin{bmatrix} \wvec_1 & \cdots & \wvec_q \end{bmatrix}\) is a basis of \(S\text{.}\) Then we write
\begin{equation*}
[T]^\bcal_\ccal = \bigl[ \begin{array}{ccc} [T(\vvec_1)]_\ccal & \cdots
& [T(\vvec_p)]_\ccal \end{array} \bigr] .
\end{equation*}
Proposition 2.7.2.
The matrix \([T]^\bcal_\ccal\) has the property that
\begin{equation*}
[T]^\bcal_\ccal [\xvec]_\bcal = [T(\xvec)]_\ccal
\end{equation*}
for every vector \(\xvec\) in \(R\text{.}\)
If \(\bcal\) and \(\ccal\) are bases of the same subspace \(S\text{,}\) and \(I\) represents the identity transformation \(I(\xvec) = \xvec\) then \([I]^\bcal_\ccal\) is the change of basis matrix from basis \(\bcal\) to basis \(\ccal\text{.}\)
Subsection 2.7.2 Choosing convenient bases
Sometimes we can find matrix representations of linear transformations by first finding a convenient basis where the transformation is easier to describe and then changing coordinates to a desired coordinate system.
Activity 2.7.1. Projection matrix.
Suppose that \(T : \real^2 \to \real^2 \) is the orthogonal projection of \(\real^2 \) onto the line spanned by \(\begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{.}\) Let \(\bcal = \begin{bmatrix} \vvec_1 & \vvec_2 \end{bmatrix} \) be the basis of \(\real^2 \) consisting of the vectors \(\vvec_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \) and \(\vvec_2 = \begin{bmatrix} 2 \\ -1 \end{bmatrix} \text{.}\)
-
Find the change of basis matrices \([I]^\bcal_\ecal\) and \([I]^\ecal_\bcal\) where \(\ecal = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \) is the standard basis.
-
Combine your calculations above to find \([T]^\ecal_\ecal\text{,}\) the matrix of \(T\) in the standard basis.
Subsection 2.7.3 Injectivity and surjectivity
Exercises 2.7.4 Exercises
1.
Let \(T : \real^2 \to \real^2\) be the linear transformation that reflects the plane across the line through the origin with slope \(\frac 32\text{.}\) Find a matrix \(A\) such that \(T(\xvec) = A \xvec\) for every vector \(\xvec\) in \(\real^2\text{.}\)
2.
Let \(T : \real^2 \to \real^2\) be the linear transformation that rotates the plane counterclockwise by \(60^\circ\text{.}\) Let \(\bcal = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\text{.}\) Compute \([T]^\bcal_\bcal\text{.}\)
3.
Let \(T : \real^3 \to \real^3\) be the orthogonal projection on the plane with equation \(x + y + z = 0\text{.}\) Find a matrix \(A\) such that \(T(\xvec) = A \xvec\) for every \(\xvec\) in \(\real^3\text{.}\) Hints : first find a basis \(\bcal\) such that \([T]^\bcal_\bcal\) is easier to calculate. Then use change of basis to find \([T]^\ecal_\ecal\text{.}\)
Hint.
4.
Let \(T : \real^3 \to \real^3\) be the linear transformation that rotates space counterclockwise by \(60^\circ\) around the vector \(\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\text{.}\) Find the matrix of \(T\) in the standard basis.
Hint 2.
5.
If \(A\) is a \(2 \times 2\) matrix that represents reflection across the \(x\)-axis, and \(B\) is a \(2 \times 2\) matrix that represents a counterclockwise rotation by \(60^\circ = \frac\pi3\) around the origin, what geometric operation will \(B A B^{-1}\) represent ? What geometric operation will \(A B A^{-1}\) represent ?
