1 Show that the matrices $\left[\begin{array}{ll}1 & 0 \ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \ 1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \ 0 & 1\end{array}\right]$ are linearly independent elements of $M_{2,2}(\mathbb{R})$, and hence that they form a basis for $M_{2,2}(\mathbb{R}) .$
2 Check whether the following are linearly dependent or independent:
(a) $(1,0,-1,1),(0,1,-3,2),(-1,2,0,1),(0,4,0,-1)$ in $\mathbb{R}^{4}$;
(b) $1-x, 1+x, 1-x+2 x^{2}$ in $P_{2}$;
(c) $1+i, 1-i, 2+3 i$ in $\mathbb{C}$ (as a vector space over $\mathbb{R}$ );
GUIDE TO LINEAR ALGEBRA
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(d) $\left[\begin{array}{rr}1 & -1 \ -1 & 1\end{array}\right],\left[\begin{array}{rr}-1 & 1 \ 1 & -1\end{array}\right],\left[\begin{array}{ll}1 & 2 \ 3 & 4\end{array}\right],\left[\begin{array}{rr}0 & -3 \ -4 & -3\end{array}\right]$ in $M_{2,2}(\mathbb{R})$.

3 Show that each of the following sequences form a basis for the given vector space:
(a) $(1,-2,-1),(-1,0,3),(0,2,-1)$ for $\mathbb{R}^{3}$;
(b) $3+x^{3}, 2-x-x^{2}, x+x^{2}-x^{3}, x+2 x^{2}$ for $P_{3}$;
(c) $1+i, 1-i$ for $\mathbb{C}$ (as a vector space over $\mathrm{R}$ );
(d) $\left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}\right],\left[\begin{array}{ll}0 & 1 \ 1 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \ 1 & 0\end{array}\right],\left[\begin{array}{rr}0 & 1 \ 1 & -1\end{array}\right]$ for $M_{2,2}(\mathbb{R})$.
4 Write $(1,4,-5)$ as a linear combination of the basis vectors in 3(a) above.
5 Write $1+2 x+3 x^{2}+4 x^{3}$ as a linear combination of the basis vectors in 3 (b) above.
6 Prove Lemma 5.3.2. (If you have difficulties look back at section 1.2.)
7 Prove that any sequence of vectors in a vector space $V$ which includes the zero vector must be linearly dependent.
8 Prove that, in any vector space $V$, a sequence of vectors in which the same vector occurs more than once is necessarily linearly dependent.

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