1. ## span

1. Find three distinct non-zero vectors $u,v,w$ in $\mathbb{R}^{3}$ such that $\text{span}( \{u,v \}) = \text{span}( \{v,w \}) = \text{span}( \{u,v,w \})$ but such that $\text{span}(\{u,w \}) \neq \text{span}(\{u,v,w \})$.

Consider $u = (1,0,0)$, $v = (0,1,0)$, and $w = (1/2,0,0)$. Then $\text{span}( \{u,v \}) = \text{span}( \{v,w \}) = \text{span}( \{u,v,w \})$. But $\text{span}(\{u,w \}) \neq \text{span}(\{u,v,w \})$.

2. Find a basis for $M_{2 \times 2}^{0} (\mathbb{R})$, the vector space of $2 \times 2$ matrices with trace zero. Why is this set a basis?

So we want to find a spanning set that is linearly independent. Consider $B = \left \{ \begin{bmatrix} 1 & a \\ b & -1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \right \}$ where $a$ and $b$ can be chosen freely. This is a basis because it is a linearly independent set that spans $M_{2 \times 2}^{0}(\mathbb{R})$.

Is this correct?

2. Part 1 looks good to me, but for number two, your second basis element is not even in the set $Tr(I_2)=1+1=2 \not = 0$.
I think these are the ones you want.
$\left[\begin{array}{cc}1 & 0 \\0 & -1\end{array}\right]$
$\left[\begin{array}{cc}0 & 1 \\0 & 0\end{array}\right]$
$\left[\begin{array}{cc}0 & 0 \\1 & 0\end{array}\right]$

Linear independence and spanning are both pretty clear.