More on Linear Transformations#
Table of Contents#
In the following, letters that designate transformations and matrices are arbitrary and could be replaced by the letters \(T\) and \(\mathbf{A}\) respectively.
Reflection#
Reflection about the \(y\)-axis in \(\mathbb{R}^2\)#
The linear transformation \(A : \mathbb{R}^2 \to \mathbb{R}^2\) that reflects points about the \(y\)-axis.
\( A \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xA \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) + yA \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) = x \begin{bmatrix*}[r] -1 \\ 0 \\ \end{bmatrix*} + y \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} = \begin{bmatrix*}[r] -x \\ y \\ \end{bmatrix*} \)
\( A \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{A}}{ \begin{bmatrix*}[r] -1 & 0 \\ 0 & 1 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((-x, y)\) when reflected about the \(y\)-axis.
Reflection about the \(x\)-axis in \(\mathbb{R}^2\)#
The linear transformation \(B : \mathbb{R}^2 \to \mathbb{R}^2\) that reflects points about the \(x\)-axis.
\( B \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xB \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yB \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] 0 \\ -1 \\ \end{bmatrix*} } = \begin{bmatrix*}[r] x \\ -y \\ \end{bmatrix*} \)
\( B \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{B}}{ \begin{bmatrix*}[r] 1 & 0 \\ 0 & -1 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((x, -y)\) when reflected about the \(x\)-axis.
Reflection about the line \(y = x\) in \(\mathbb{R}^2\)#
The linear transformation \(C : \mathbb{R}^2 \to \mathbb{R}^2\) that reflects points about the line \(y = x\).
\( C \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xC \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yC \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} } = \begin{bmatrix*}[r] y \\ x \\ \end{bmatrix*} \)
\( C \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{C}}{ \begin{bmatrix*}[r] 0 & 1 \\ 1 & 0 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((y, x)\) when reflected about the line \(y = x\).
Reflection about the line \(y = -x\) in \(\mathbb{R}^2\)#
The linear transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) that reflects points about the line \(y = x\).
\( T \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xT \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yT \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] 0 \\ -1 \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] -1 \\ 0 \\ \end{bmatrix*} } = \begin{bmatrix*}[r] -y \\ -x \\ \end{bmatrix*} \)
\( T \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{A}}{ \begin{bmatrix*}[r] 0 & -1 \\ -1 & 0 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((-y, -x)\) when reflected about the line \(y = -x\).
Reflection about the origin#
The linear transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) that reflects points about the origin.
\( T \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xT \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yT \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] -1 \\ 0 \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] 0 \\ -1 \\ \end{bmatrix*} } = \begin{bmatrix*}[r] -x \\ -y \\ \end{bmatrix*} \)
\( T \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{A}}{ \begin{bmatrix*}[r] -1 & 0 \\ 0 & -1 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((-x, -y)\) when reflected about the origin.
Reflection first about the \(x\)-axis and then about the line \(y = x\) in \(\mathbb{R}^2\)#
The linear transformation \(D : \mathbb{R}^2 \to \mathbb{R}^2\) that first reflects points about the \(x\)-axis and then reflects points about the line \(y = x\).
\( D \left ( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = C \left ( B \left ( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) \right) = C \left ( \underset{\mathbf{B}}{ \begin{bmatrix*}[r] 1 & 0 \\ 0 & -1 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \right) = \underset{\mathbf{C}}{ \begin{bmatrix*}[r] 0 & 1 \\ 1 & 0 \\ \end {bmatrix*} } \underset{\mathbf{B}}{ \begin{bmatrix*}[r] 1 & 0 \\ 0 & -1 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } = \underset{\mathbf{D}}{ \begin{bmatrix*}[r] 0 & -1 \\ 1 & 0 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
\( \begin{aligned} & \underset{\mathbf{B}}{ \begin{bmatrix*}[r] 1 & 0 \\ 0 & -1 \\ \end {bmatrix*}} \\ \underset{\mathbf{C}}{ \begin{bmatrix*}[r] 0 & 1 \\ 1 & 0 \\ \end {bmatrix*}} & \begin{bmatrix*}[r] 0 \cdot 1 + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot -1 \\ 1 \cdot 1 + 0 \cdot 0 & 1 \cdot 0 + 0 \cdot -1 \\ \end {bmatrix*} \\ \end {aligned} \\ \begin{aligned} & \underset{\mathbf{B}}{ \begin{bmatrix*}[r] 1 & 0 \\ 0 & -1 \\ \end {bmatrix*}} \\ \underset{\mathbf{C}}{ \begin{bmatrix*}[r] 0 & 1 \\ 1 & 0 \\ \end {bmatrix*}} & \underset{\mathbf{D}}{ \begin{bmatrix*}[r] 0 & -1 \\ 1 & 0 \\ \end {bmatrix*}} \\ \end {aligned} \)
Thus a point \((x, y)\) becomes \((-y, x)\) when first reflected about the \(x\)-axis and then reflected about the line \(y = x\).
Contraction and expansion#
Horizontal contraction and expansion in \(\mathbb{R}^2\)#
The linear transformation \(F : \mathbb{R}^2 \to \mathbb{R}^2\) that represents any horizontal contraction or expansion.
\( F \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xF \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yF \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] k \\ 0 \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} } = \begin{bmatrix*}[r] kx \\ y \\ \end{bmatrix*} \)
\( F \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{F}}{ \begin{bmatrix*}[r] k & 0 \\ 0 & 1 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((kx, y)\) with \(0 \lt k \lt 1\) when horizontally contracted and with \(k \gt 1\) when horizontally expanded.
Vertical contraction and expansion in \(\mathbb{R}^2\)#
The linear transformation \(F : \mathbb{R}^2 \to \mathbb{R}^2\) that represents any vertical contraction or expansion.
\( G \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xG \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yG \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] 0 \\ k \\ \end{bmatrix*} } = \begin{bmatrix*}[r] x \\ ky \\ \end{bmatrix*} \)
\( G \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{G}}{ \begin{bmatrix*}[r] 1 & 0 \\ 0 & k \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((x, ky)\) with \(0 \lt k \lt 1\) when vertically contracted and with \(k \gt 1\) when vertically expanded.
Projection#
Projection onto the \(x\)-axis in \(\mathbb{R}^2\)#
The linear transformation \(P : \mathbb{R}^2 \to \mathbb{R}^2\) that represents projection onto the \(x\)-axis.
\( P \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xP \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yP \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] 0 \\ 0 \\ \end{bmatrix*} } = \begin{bmatrix*}[r] x \\ 0 \\ \end{bmatrix*} \)
\( P \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{P}}{ \begin{bmatrix*}[r] 1 & 0 \\ 0 & 0 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((x, 0)\) when projected onto the \(x\)-axis.
Projection onto the \(y\)-axis in \(\mathbb{R}^2\)#
The linear transformation \(Q : \mathbb{R}^2 \to \mathbb{R}^2\) that represents projection onto the \(y\)-axis.
\( Q \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xQ \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yQ \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] 0 \\ 0 \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} } = \begin{bmatrix*}[r] 0 \\ y \\ \end{bmatrix*} \)
\( Q \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{Q}}{ \begin{bmatrix*}[r] 0 & 0 \\ 0 & 1 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((0, y)\) when projected onto the \(y\)-axis.
Rotation#
Rotation by an angle \(\theta\) in the counterclockwise direction about the origin in \(\mathbb{R}^2\)#
The linear transformation \(R : \mathbb{R}^2 \to \mathbb{R}^2\) that represents rotation by an angle \(\theta\) in the counterclockwise direction about the origin.
\( R \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xR \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yR \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] \cos \theta \\ \sin \theta \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] -\sin \theta \\ \cos \theta \\ \end{bmatrix*} } = \begin{bmatrix*}[r] x \cos \theta - y \sin \theta \\ -x \sin \theta + y \cos \theta \\ \end {bmatrix*} \)
\( R \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{R}}{ \begin{bmatrix*}[r] \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Shear#
Horizontal shear in \(\mathbb{R}^2\)#
The linear transformation \(S : \mathbb{R}^2 \to \mathbb{R}^2\) that represents any horizontal shear.
\( S \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xS \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yS \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] k \\ 1 \\ \end{bmatrix*} } = \begin{bmatrix*}[r] x + ky \\ y \\ \end{bmatrix*} \)
\( S \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{S}}{ \begin{bmatrix*}[r] 1 & k \\ 0 & 1 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((x + ky, y)\) with \(k \lt 0\) when sheared to the left and with \(k \gt 0\) when sheared to the right.
Vertical shear in \(\mathbb{R}^2\)#
The linear transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) that represents any vertical shear.
\( S \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = xS \underset{\mathbf{e_1}}{ \left( \begin{bmatrix*}[r] 1 \\ 0 \\ \end{bmatrix*} \right) } + yS \underset{\mathbf{e_2}}{ \left( \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} \right) } = x \underset{\mathbf{u_1}}{ \begin{bmatrix*}[r] 1 \\ k \\ \end{bmatrix*} } + y \underset{\mathbf{u_2}}{ \begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*} } = \begin{bmatrix*}[r] x \\ kx + y \\ \end{bmatrix*} \)
\( S \left( \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} \right) = \underset{\mathbf{S}}{ \begin{bmatrix*}[r] 1 & 0 \\ k & 1 \\ \end {bmatrix*} } \underset{\mathbf{x}}{ \begin{bmatrix*}[r] x \\ y \\ \end{bmatrix*} } \)
Thus a point \((x, y)\) becomes \((x, kx + y)\) with \(k \lt 0\) when sheared down and with \(k \gt 0\) when sheared up.