Exercise 1

Decide if $\mathbb{R}^{2}$ is the direct sum of each $W_{1}$ and $W_{2}$.

• $W_{1}=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ 0\end{pmatrix}\mid x\in\mathbb{R}\hskip 1.00006pt\}$}$, $W_{2}=\text{$\{\hskip 1.00006pt\begin{pmatrix}x\\ x\end{pmatrix}\mid x\in\mathbb{R}\hskip 1.00006pt\}$}$

• $W_{1}=\text{$\{\hskip 1.00006pt\begin{pmatrix}s\\ s\end{pmatrix}\mid s\in\mathbb{R}\hskip 1.00006pt\}$}$, $W_{2}=\text{$\{\hskip 1.00006pt\begin{pmatrix}s\\ 1.1s\end{pmatrix}\mid s\in\mathbb{R}\hskip 1.00006pt\}$}$

• $W_{1}=\mathbb{R}^{2}$, $W_{2}=\text{$\{\hskip 1.00006pt\vec{0}\hskip 1.00006pt\}$}$

• $W_{1}=W_{2}=\text{$\{\hskip 1.00006pt\begin{pmatrix}t\\ t\end{pmatrix}\mid t\in\mathbb{R}\hskip 1.00006pt\}$}$

• $W_{1}=\text{$\{\hskip 1.00006pt\begin{pmatrix}1\\ 0\end{pmatrix}+\begin{pmatrix}x\\ 0\end{pmatrix}\mid x\in\mathbb{R}\hskip 1.00006pt\}$}$, $W_{2}=\text{$\{\hskip 1.00006pt\begin{pmatrix}-1\\ 0\end{pmatrix}+\begin{pmatrix}0\\ y\end{pmatrix}\mid y\in\mathbb{R}\hskip 1.00006pt\}$}$