Q1: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be a linear map given by $f(x_1, x_2) = \begin{pmatrix} 1 & 2 \ 2 & -1 \ -3 & 0 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \end{pmatrix}$ and $g: \mathbb{R}^3 \rightarrow \mathbb{R}$ given by $g(x_1, x_2, x_3) = x_1 + x_2 + x_3$. What is the composition $ g \circ f$?

A1: It’s a linear map $g \circ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ given by $(g \circ f)(x_1, x_2) = x_2 $

A2: It’s a linear map $g \circ f: \mathbb{R}^3 \rightarrow \mathbb{R}^2 $ given by $(g \circ f)(x_1, x_2, x_3) = \begin{pmatrix} x_1 \ x_2 \end{pmatrix} $

A3: It’s a linear map $g \circ f: \mathbb{R} \rightarrow \mathbb{R} $ given by $(g \circ f)(x) = x $

A4: It’s not a linear map.

Q2: Consider the set of linear maps between two $\mathbb{F}$-vector spaces $V$ and $W$, both of finite dimension, denoted by $\mathcal{L}(V,W)$. What is correct?

A1: $\mathcal{L}(V,W)$ is an $\mathbb{F}$-vector space of dimension $\dim(V) \cdot \dim(W)$.

A2: $\mathcal{L}(V,W)$ is not a vector space.

A3: $\mathcal{L}(V,W)$ is an $\mathbb{F}$-vector space of dimension $\dim(V) + \dim(W)$.

A4: $\mathcal{L}(V,W)$ is the empty set.