4.2 Exercises

20 points · 4 extra credit

Null Spaces, Column Spaces, and Linear Transformations

1. 1. (3 pts) Find a basis for $\operatorname{Nul} A$, where $A = \begin{bmatrix} 1 & 2 & -1 & 3 \\ 2 & 4 & -1 & 8 \\ 1 & 2 & 0 & 5 \end{bmatrix}.$

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2. 2. (3 pts) Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \end{bmatrix}$. Describe $\operatorname{Col} A$, and determine whether each of $\mathbf{b}_1 = (2, 4)$ and $\mathbf{b}_2 = (1, 3)$ lies in $\operatorname{Col} A$.

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3. 3. (3 pts) Define $T : P_2 \to \mathbb{R}^2$ by $T(p) = (p(0),\, p(1))$.

   (a) Show that $T$ is a linear transformation.

   (b) Find a basis for the kernel of $T$.

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4. 4. (2 pts) Let $A$ be a $3\times 5$ matrix. State which space ($\mathbb{R}^3$ or $\mathbb{R}^5$) contains $\operatorname{Nul} A$ and which contains $\operatorname{Col} A$, and explain why.

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5. 5. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) $\operatorname{Nul} A$ is the set of all solutions of $A\mathbf{x} = \mathbf{0}$.

   (b) $\operatorname{Col} A$ is the set of all $\mathbf{b}$ for which $A\mathbf{x} = \mathbf{b}$ is consistent.

   (c) If $\operatorname{Col} A = \mathbb{R}^m$ for an $m\times n$ matrix $A$, then $A\mathbf{x} = \mathbf{b}$ has a solution for every $\mathbf{b}$ in $\mathbb{R}^m$.

   (d) The kernel of a linear transformation $T : V \to W$ is a subspace of the codomain $W$.

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6. 6. (3 pts) Let $A$ be an $m\times n$ matrix. Prove that $\operatorname{Nul} A$ is a subspace of $\mathbb{R}^n$.

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7. 7. (3 pts) Find a basis for $\operatorname{Col} A$, where $A = \begin{bmatrix} 1 & 2 & 0 & 1 \\ 2 & 4 & 1 & 3 \\ 3 & 6 & 1 & 4 \end{bmatrix}.$

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8. 8. (4 pts ••) Let $T : V \to W$ be a linear transformation. Prove that $T(\mathbf{0}_V) = \mathbf{0}_W$, and prove that the range of $T$ is a subspace of $W$.

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