If a linear transformation $f : V \to W$ is given by $f(\mathbf{x}) = A \mathbf{x}$ for some $A \in M_{n,k}(\mathbb{R})$, then:

1. the domain of $f$ is $V =$ choose R^k R^n ,
2. the codomain of $f$ is $W =$ choose R^k R^n ,
3. the kernel of $f$ is equal to the choose column space null space row space subspace of $A$, and
4. the image of $f$ is equal to the choose column space null space row space subspace of $A$.

Suppose $A \in M_{n,k}(\mathbb{R})$. Write $A = (\mathbf{a_1} \mid \cdots \mid \mathbf{a_k})$.

1. A vector $\mathbf{x} \in \mathbb{R}^k$ is in the null space of $A$ if (select all that apply):
   
   A. $A \mathbf{x} = \mathbf{0}_n$.
   B. $\mathrm{rref}(A \mid \mathbf{y})$ has no rows of the form $(0 \ldots 0 \mid 1)$.
   C. The dot product of $\mathbf{x}$ with every row vector from $A$ is $0$.
   D. $x_1 \mathbf{a_1} + \cdots + x_k \mathbf{a_k} = \mathbf{0}_n$
   E. $\mathbf{y} = x_1 \mathbf{a_1} + \cdots + x_k \mathbf{a_k}$ for some $\langle x_1, \ldots, x_k \rangle \in \mathbb{R}^k$.
   F. $\mathbf{y} = A \mathbf{x}$ for some $\mathbf{x} \in \mathbb{R}^k$.
   G. $\mathbf{y} \in \mathrm{span} \lbrace \mathbf{a_1}, \ldots, \mathbf{a_k} \rbrace$.
   
2. A vector $\mathbf{y} \in \mathbb{R}^n$ is in the column space of $A$ if (select all that apply):
   
   A. $\mathrm{rref}(A \mid \mathbf{y})$ has no rows of the form $(0 \ldots 0 \mid 1)$.
   B. $\mathbf{y} \in \mathrm{span} \lbrace \mathbf{a_1}, \ldots, \mathbf{a_k} \rbrace$.
   C. $\mathbf{y} = x_1 \mathbf{a_1} + \cdots + x_k \mathbf{a_k}$ for some $\langle x_1, \ldots, x_k \rangle \in \mathbb{R}^k$.
   D. $A \mathbf{x} = \mathbf{0}_n$.
   E. $x_1 \mathbf{a_1} + \cdots + x_k \mathbf{a_k} = \mathbf{0}_n$
   F. The dot product of $\mathbf{x}$ with every row vector from $A$ is $0$.
   G. $\mathbf{y} = A \mathbf{x}$ for some $\mathbf{x} \in \mathbb{R}^k$.
   

Suppose $f$ is the function defined by $f(\mathbf{x}) = A \mathbf{x}$, where

1. The domain of $f$ is . For instance, enter R^5 for $\mathbb{R}^5$.
2. The codomain of $f$ is . For instance, enter R^5 for $\mathbb{R}^5$.
3. Select all of the vectors that are in the kernel of $f$. You should be able to justify your answers (there may be more than one correct answer).
   
   A. $<-3,2,-3>$
   B. $<3,-3,0>$
   C. $<0,1,0>$
   D. $<0,-2,1,1>$
   E. $<0,0,1,0>$
   F. $<-4,0,1,1>$
   
4. Select all of the vectors that are in the image of $f$. You should be able to justify your answers (there may be more than one correct answer).
   
   A. $<0,0,1,0>$
   B. $<-4,0,1,1>$
   C. $<3,-3,0>$
   D. $<0,1,0>$
   E. $<-3,2,-3>$
   F. $<0,-2,1,1>$