1.5.E: Linear and Affine Functions (Exercises)

Exercise \(\PageIndex{1}\)

Let \(\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}\) be vectors in \(\mathbb{R}^{m}\) and define \(L: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\) by

\[ L(\mathbf{x})=\left(\mathbf{a}_{1} \cdot \mathbf{x}, \mathbf{a}_{2} \cdot \mathbf{x}, \ldots, \mathbf{a}_{n} \cdot \mathbf{x}\right). \nonumber \]

Show that \(L\) is linear. What does \(L\) look like in the special cases

1. \(m=n=1 ?\)
2. \(n=1 ?\)
3. \(m=1 ?\)

Exercise \(\PageIndex{2}\)

For each of the following functions \(f\), find the dimension of the domain space, the dimension of the range space, and state whether the function is linear, affine, or neither.

1. \(f(x, y)=(3 x-y, 4 x, x+y)\)
2. \(f(x, y)=(4 x+7 y, 5 x y)\)
3. \(f(x, y, z)=(3 x+z, y-z, y-2 x)\)
4. \(f(x, y, z)=(3 x-4 z, x+y+2 z)\)
5. \(f(x, y, z)=\left(3 x+5, y+z, \frac{1}{x+y+z}\right)\)
6. \(f(x, y)=3 x+y-2\)
7. \(f(x)=(x, 3 x)\)
8. \(f(w, x, y, z)=(3 x, w+x-y+z-5)\)
9. \(f(x, y)=(\sin (x+y), x+y)\)
10. \(f(x, y)=\left(x^{2}+y^{2}, x-y, x^{2}-y^{2}\right)\)
11. \(f(x, y, z)=(3 x+5, y+z, 3 x-z+6, z-1)\)

Answer

(a) Dimension of the domain space = 2; dimension of the range space = 3; \(f\) is linear

(b) Dimension of the domain space = 2; dimension of the range space = 2; \(f\) is neither linear nor affine

(c) Dimension of the domain space = 3; dimension of the range space = 3; \(f\) is linear

(d) Dimension of the domain space = 3; dimension of the range space = 2; \(f\) is linear

(e) Dimension of the domain space = 3; dimension of the range space = 4; \(f\) is affine

(f) Dimension of the domain space = 2; dimension of the range space = 1; \(f\) is affine

(g) Dimension of the domain space = 1; dimension of the range space = 2; \(f\) is linear

(h) Dimension of the domain space = 4; dimension of the range space = 2; \(f\) is linear

(i) Dimension of the domain space = 2; dimension of the range space = 2; \(f\) is neither linear nor affine

(j) Dimension of the domain space = 2; dimension of the range space = 3; \(f\) is neither linear nor affine

Exercise \(\PageIndex{3}\)

For each of the following linear functions \(L\), find a matrix \(M\) such that \(L(\mathbf{x})=M \mathbf{x}\).

1. \(L(x, y)=(x+y, 2 x-3 y)\)
2. \(L(w, x, y, z)=(x, y, z, w)\)
3. \(L(x)=(3 x, x, 4 x)\)
4. \(L(x)=-5 x\)
5. \(L(x, y, z)=4 x-3 y+2 z\)
6. \(L(x, y, z)=(x+y+z, 3 x-y, y+2 z)\)
7. \(L(x, y)=(2 x, 3 y, x+y, x-y, 2 x-3 y)\)
8. \(L(x, y)=(x, y)\)
9. \(L(w, x, y, z)=(2 w+x-y+3 z, w+2 x-3 z)\)

Answer

(a) \(M=\left[\begin{array}{rr}
1 & 1 \\
2 & -3
\end{array}\right]\)

(b) \(M=\left[\begin{array}{rrrr}
2 & 1 & -1 & 3 \\
1 & 2 & 0 & -3
\end{array}\right]\)

(c) \(M=\left[\begin{array}{l}
3 \\
1 \\
4
\end{array}\right]\)

(d) \(M=[-5]\)

(e) \(M=\left[\begin{array}{lll}
4 & -3 & 2
\end{array}\right]\)

(f) \(M=\left[\begin{array}{rrr}
1 & 1 & 1 \\
3 & -1 & 0 \\
0 & 1 & 2
\end{array}\right]\)

(g) \(M=\left[\begin{array}{rr}
2 & 0 \\
0 & 3 \\
1 & 1 \\
1 & -1 \\
2 & -3
\end{array}\right]\)

(h) \(M=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\)

(i) \(M=\left[\begin{array}{rrrr}
2 & 1 & -1 & 3 \\
1 & 2 & 0 & -3
\end{array}\right]\)

Exercise \(\PageIndex{4}\)

For each of the following affine functions \(A\), find a matrix \(M\) and a vector \(\mathbf{b}\) such that \(A(\mathbf{x})=M \mathbf{x}+\mathbf{b}\).

1. \(A(x, y)=(3 x+4 y-6,2 x+y-3)\)
2. \(A(x)=3 x-4\)
3. \(A(x, y, z)=(3 x+y-4, y-z+1,5)\)
4. \(A(w, x, y, z)=(1,2,3,4)\)
5. \(A(x, y, z)=3 x-4 y+z-1\)
6. \(A(x)=(3 x,-x, 2)\)
7. \(A\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}-x_{2}+1, x_{1}-x_{3}+1, x_{2}+x_{3}\right)\)

Exercise \(\PageIndex{5}\)

Multiply the following.

(a) \( \left[\begin{array}{lll}
1 & 2 & 3 \\
3 & 2 & 1
\end{array}\right]\left[\begin{array}{r}
1 \\
2 \\
-3
\end{array}\right] \)

(b) \(\left[\begin{array}{rr}
-1 & 2 \\
3 & -2 \\
-1 & 1
\end{array}\right]\left[\begin{array}{r}
3 \\
-1
\end{array}\right]\)

(c) \(\left[\begin{array}{lll}
1 & 2 & 1-3
\end{array}\right]\left[\begin{array}{r}
2 \\
3 \\
-2 \\
1
\end{array}\right]\)

(d) \(\left[\begin{array}{lll}
1 & 2 & 1 \\
3 & 2 & 3 \\
0 & 1 & 2
\end{array}\right]\left[\begin{array}{r}
2 \\
-1 \\
2
\end{array}\right]\)

Answer

(a) \(\left[\begin{array}{r}
-4 \\
4
\end{array}\right]\)

(b) \(\left[\begin{array}{c}
-5 \\
11 \\
-4
\end{array}\right]\)

(c) \([3]\)

(d) \(\left[\begin{array}{c}
2 \\
10 \\
3
\end{array}\right]\)

Exercise \(\PageIndex{6}\)

Let \(L: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be the linear function that maps a vector \(\mathbf{x}=(x, y)\) to its reflection across the horizontal axis. Find the matrix \(M\) such that \(L(\mathbf{x})=M \mathbf{x}\) for all \(\mathbf{x}\) in \(\mathbb{R}^{2}\).

Exercise \(\PageIndex{7}\)

Let \(L: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be the linear function that maps a vector \(\mathbf{x}=(x, y)\) to its reflection across the line \(y=x\). Find the matrix \(M\) such that \(L(\mathbf{x})=M \mathbf{x}\) for all \(\mathbf{x}\) in \(\mathbb{R}^{2}\).

Answer

\(M=\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]\)

Exercise \(\PageIndex{8}\)

Let \(L: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be the linear function that maps a vector \(\mathbf{x}=(x, y)\) to its reflection across the line \(y=-x\). Find the matrix \(M\) such that \(L(\mathbf{x})=M \mathbf{x}\) for all \(\mathbf{x}\) in \(\mathbb{R}^{2}\).

Answer

\(M=\left[\begin{array}{rr}
0 & -1 \\
-1 & 0
\end{array}\right]\)

Exercise \(\PageIndex{9}\)

Let \(R_{\theta}\) be defined as in (1.5.12).

1. Show that for any \(\mathbf{x}\) in \(\mathbb{R}^{2}\), \(\left\|R_{\theta}(\mathbf{x})\right\|=\|\mathbf{x}\| \).
2. For any \(\mathbf{x}\) in \(\mathbb{R}^{2}\), let \( \alpha \) be the angle between \(\mathbf{x}\) and \(R_{\theta}(\mathbf{x})\). Show that \(\cos (\alpha)= \cos(\theta)\). Together with (a), this verifies that \(R_{\theta}(\mathbf{x})\) is the rotation of \(\mathbf{x}\) through an angle θ.

Exercise \(\PageIndex{10}\)

Let \(S_{\theta}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be the linear function that rotates a vector \(\mathbf{x}\) clockwise through an angle \(\theta\). Find the matrix \(M\) such that \(S_{\theta}(\mathbf{x})=M \mathbf{x}\) for all \(\mathbf{x}\) in \(\mathbb{R}^{2}\).

Answer

\(M=\left[\begin{array}{rr}
\cos (\theta) & \sin (\theta) \\
-\sin (\theta) & \cos (\theta)
\end{array}\right]\)

Exercise \(\PageIndex{11}\)

Given a function \(f: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\), we call the set

\[ \left\{\mathbf{y}: \mathbf{y}=f(\mathbf{x}) \text { for some } \mathbf{x} \text { in } \mathbb{R}^{m}\right\} \nonumber \]

the image, or range, of \(f\).

1. Suppose \(L: \mathbb{R} \rightarrow \mathbb{R}^{n}\) is linear with \(L(1) \neq \mathbf{0}\). Show that the image of \(L\) is a line in \(\mathbb{R}^{n}\) which passes through \(\mathbf{0}\).
2. Suppose \(L: \mathbb{R}^2 \rightarrow \mathbb{R}^{n}\) is linear and \(L\left(\mathbf{e}_{1}\right)\) and \(L\left(\mathbf{e}_{2}\right)\) are linearly independent. Show that the image of \(L\) is a plane in \(\mathbb{R}^{n}\) which passes through \(\mathbf{0}\).

Exercise \(\PageIndex{12}\)

Given a function \(f: \mathbb{R}^{m} \rightarrow \mathbb{R}\), we call the set

\[ \left\{\left(x_{1}, x_{2}, \ldots, x_{m}, x_{m+1}\right): x_{m+1}=f\left(x_{1}, x_{2}, \ldots, x_{m}\right)\right\} \nonumber \]

the graph of \(f\). Show that if \(L: \mathbb{R}^{m} \rightarrow \mathbb{R}\) is linear, then the graph of \(L\) is a hyperplane in \(\mathbb{R}^{m+1}\).