2.8 Exercises

20 points · 4 extra credit

Subspaces of $\mathbb{R}^n$

Problems 2–4 and 7 use the matrix

$A = \begin{bmatrix} 1 & 3 & 2 & 4 \\ 2 & 6 & 5 & 7 \\ 1 & 3 & 1 & 5 \end{bmatrix}.$

1. (3 pts) Determine whether each set is a subspace of $\mathbb{R}^3$ . Justify each answer.

(a) $H_1 = \{(x, y, z) : x + 2y - z = 0\}$ .

(b) $H_2 = \{(x, y, z) : x \ge 0\}$ .

(c) $H_3 = \{(x, y, z) : x + y + z = 1\}$ .
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2. (3 pts) Find a basis for $\operatorname{Col} A$ .
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3. (3 pts) Find a basis for $\operatorname{Nul} A$ .
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4. (2 pts) Determine whether $\mathbf{b} = \begin{bmatrix} 3 \\ 7 \\ 2 \end{bmatrix}$ is in $\operatorname{Col} A$ .
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5. (3 pts) State whether each is true or false, and justify. Let $A$ be an $m\times n$ matrix.

(a) $\operatorname{Nul} A$ is a subspace of $\mathbb{R}^m$ .

(b) $\operatorname{Col} A$ is a subspace of $\mathbb{R}^m$ .

(c) A single nonzero vector $\mathbf{v}$ in $\mathbb{R}^n$ spans a subspace (the line through the origin in the direction of $\mathbf{v}$ ).
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6. (3 pts) Prove directly from the definition of a subspace that $\operatorname{Nul} A$ is a subspace of $\mathbb{R}^n$ for any $m\times n$ matrix $A$ .
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7. (3 pts) Determine whether $\mathbf{v} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ is in $\operatorname{Col} A$ , and explain what your answer says about the solvability of $A\mathbf{x} = \mathbf{v}$ .
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8. (4 pts ••) Let $U$ and $W$ be subspaces of $\mathbb{R}^n$ . Prove that $U \cap W$ is a subspace of $\mathbb{R}^n$ , and give an explicit example showing that $U \cup W$ need not be a subspace.
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