Math Problems Vault

Problem 4.2.1

Determine spanning sets for each of the four fundamental subspaces associated with

\mathbf{A} = \begin{pmatrix} 1 & 2 & 1 & 1 & 5 \\ -2 & -4 & 0 & 4 & -2 \\ 1 & 2 & 2 & 4 & 9 \end{pmatrix}.

The four fundamental subspaces associated with $\mathbf{A}_{m \times n}$ are the range or column space $R(\mathbf{A}) = \left\{ \mathbf{A}\overrightarrow {x}\right\} \subseteq \mathbb {R}^{m}$ , the row space or left-hand range $R(\mathbf{A}^T) = \left\{ \mathbf{A}^T \overrightarrow {y}\right\} \subseteq \mathbb {R}^{n}$ , the nullspace $N(\mathbf{A}) = \left\{ \overrightarrow {x} \mid \mathbf{A}\overrightarrow {x} = \overrightarrow {0}\right\} \subseteq \mathbb {R}^{n}$ , and the left-hand nullspace $N(\mathbf{A}^T) = \left\{ \overrightarrow {y} \mid \mathbf{A}^T \overrightarrow {y} = \overrightarrow {0}\right\} \subseteq \mathbb {R}^{m}$ .

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Problem 4.2.2

(a)(b)

Consider a linear system of equations $\mathbf{A}_{m \times n} \overrightarrow {x} = \overrightarrow {b}$ .

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(a)

Explain why $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ is consistent if and only if $\overrightarrow {b} \in R(\mathbf{A})$ .

(b)

Explain why a consistent system $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ has a unique solution if and only if $N(\mathbf{A}) = \left\{ \overrightarrow {0}\right\}$ .

Problem 4.2.3

(a)(b)

Suppose that $\mathbf{A}$ is a $3 \times 3$ matrix such that

\mathcal{R} = \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}\right\} \quad \text{and} \quad \mathcal{N} = \left\{ \begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix}\right\}

span $R(\mathbf{A})$ and $N(\mathbf{A})$ , respectively, and consider a linear system $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ , where $\overrightarrow {b} = \begin{pmatrix} 1 \\ -7 \\ 0 \end{pmatrix}$ .

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(a)

Explain why $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ must be consistent.

(b)

Explain why $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ cannot have a unique solution.

Problem 4.2.4

If $\mathbf{A} = \begin{pmatrix} -1 & 1 & 1 & -2 & 1 \\ -1 & 0 & 3 & -4 & 2 \\ -1 & 0 & 3 & -5 & 3 \\ -1 & 0 & 3 & -6 & 4 \\ -1 & 0 & 3 & -6 & 4 \end{pmatrix}$ and $\overrightarrow {b} = \begin{pmatrix} -2 \\ -5 \\ -6 \\ -7 \\ -7 \end{pmatrix}$ , is $\overrightarrow {b} \in R(\mathbf{A})$ ?

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Problem 4.2.5

(a)(b)

Suppose that $\mathbf{A}$ is an $n \times n$ matrix.

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(a)

If $R(\mathbf{A}) = \mathbb {R}^{n}$ , explain why $\mathbf{A}$ must be nonsingular.

(b)

If $\mathbf{A}$ is nonsingular, describe its four fundamental subspaces.

Problem 4.2.6

(a)(b)(c)(d)

Consider the matrices $\mathbf{A} = \begin{pmatrix} 1 & 1 & 5 \\ 2 & 0 & 6 \\ 1 & 2 & 7 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} 1 & -4 & 4 \\ 4 & -8 & 6 \\ 0 & -4 & 5 \end{pmatrix}$ .

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(a)

Do $\mathbf{A}$ and $\mathbf{B}$ have the same row space?

(b)

Do $\mathbf{A}$ and $\mathbf{B}$ have the same column space?

(c)

Do $\mathbf{A}$ and $\mathbf{B}$ have the same nullspace?

(d)

Do $\mathbf{A}$ and $\mathbf{B}$ have the same left-hand nullspace?

Problem 4.2.7

If $\mathbf{A} = \begin{pmatrix} \mathbf{A}_1 \\ \mathbf{A}_2 \end{pmatrix}$ is a square matrix such that $N(\mathbf{A}_1) = R(\mathbf{A}_2^T)$ , prove that $\mathbf{A}$ must be nonsingular.

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Problem 4.2.8

Consider a linear system of equations $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ for which $\overrightarrow {y}^T \overrightarrow {b} = 0$ for every $\overrightarrow {y} \in N(\mathbf{A}^T)$ . Explain why this means the system must be consistent.

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Problem 4.2.9

For matrices $\mathbf{A}_{m \times n}$ and $\mathbf{B}_{m \times p}$ , prove that

R(\mathbf{A} \mid \mathbf{B}) = R(\mathbf{A}) + R(\mathbf{B}).

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Problem 4.2.10

(a)(b)(c)

Let $\overrightarrow {p}$ be one particular solution of a linear system $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ .

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(a)

Explain the significance of the set

\overrightarrow {p} + N(\mathbf{A}) = \left\{ \overrightarrow {p} + \overrightarrow {h} \mid \overrightarrow {h} \in N(\mathbf{A})\right\} .

(b)

If $\operatorname {rk}\left(\mathbf{A}_{3 \times 3}\right) = 1$ , sketch a picture of $\overrightarrow {p} + N(\mathbf{A})$ in $\mathbb {R}^{3}$ .

(c)

Repeat part (b) for the case when $\operatorname {rk}\left(\mathbf{A}_{3 \times 3}\right) = 2$ .

Problem 4.2.11

Suppose that $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ is a consistent system of linear equations, and let $\overrightarrow {a} \in R(\mathbf{A}^T)$ . Prove that the inner product $\overrightarrow {a}^T \overrightarrow {x}$ is constant for all solutions to $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ .

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Problem 4.2.12

(a)(b)

For matrices such that the product $\mathbf{A}\mathbf{B}$ is defined, explain why each of the following statements is true.

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(a)

$R(\mathbf{A}\mathbf{B}) \subseteq R(\mathbf{A})$ .

(b)

$N(\mathbf{A}\mathbf{B}) \supseteq N(\mathbf{B})$ .

Problem 4.2.13

Suppose that $\mathcal{B} = \left\{ \overrightarrow {b}_1, \overrightarrow {b}_2, \ldots , \overrightarrow {b}_n\right\}$ is a spanning set for $R(\mathbf{B})$ . Prove that $\mathbf{A}(\mathcal{B}) = \left\{ \mathbf{A}\overrightarrow {b}_1, \mathbf{A}\overrightarrow {b}_2, \ldots , \mathbf{A}\overrightarrow {b}_n\right\}$ is a spanning set for $R(\mathbf{A}\mathbf{B})$ .

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