Let $l_{\mathbf{x}}$ denote the straight line through $\mathbf{x}$ with directional vector $\mathbf{u} \neq \mathbf{0}$

$$l_{\mathbf{x}}=\left\{\mathbf{y} \in \mathbb{R}^{3}: \mathbf{y}=\mathbf{x}+\lambda \mathbf{u}, \lambda \in \mathbb{R}\right\}$$

Show that $l_{\mathbf{0}}$ is a subspace of $\mathbb{R}^{3}$ and show that $l_{\mathbf{x}_{1}}=l_{\mathbf{x}_{2}} \Leftrightarrow \mathbf{x}_{\mathbf{1}}=\mathbf{x}_{2}+\lambda \mathbf{u}$ for some $\lambda \in \mathbb{R}$.

For fixed $\mathbf{u} \neq \mathbf{0}$ let $\mathcal{L}$ be the set of all the parallel straight lines $l_{\mathbf{x}}\left(\mathbf{x} \in \mathbb{R}^{3}\right)$ with directional vector $\mathbf{u}$. On $\mathcal{L}$ an addition and a scalar multiplication are defined by

$$l_{\mathbf{x}}+l_{\mathbf{y}}=l_{\mathbf{x}+\mathbf{y}}, \alpha l_{\mathbf{x}}=l_{\alpha \mathbf{x}}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^{3}, \alpha \in \mathbb{R}$$

Explain why these operations are well-defined. Show that the addition is associative and that there exists a zero vector which should be identified.

You may now assume that $\mathcal{L}$ is a vector space. If $\left\{\mathbf{u}, \mathbf{b}_{1}, \mathbf{b}_{2}\right\}$ is a basis for $\mathbb{R}^{3}$ show that $\left\{l_{\mathbf{b}_{1}}, l_{\mathbf{b}_{2}}\right\}$ is a basis for $\mathcal{L}$.

For $\mathbf{u}=(1,3,-1)^{T}$ a linear map $\Phi: \mathcal{L} \rightarrow \mathcal{L}$ is defined by

$$\Phi\left(l_{(1,-1,0)^{T}}\right)=l_{(2,4,-1)^{T}}, \Phi\left(l_{(1,1,0)^{T}}\right)=l_{(-4,-2,1)^{T}}$$

Find the matrix $A$ of $\Phi$ with respect to the basis $\left\{l_{(1,0,0)^{T}}, l_{(0,1,0)^{T}}\right\}$.