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1.\ \ Let $v_1 = (1,2),\ \ v_2=(3,0),\ \ v_3=(1,1).$
\itemitem{a)}Is $\{v_1, v_2\}$ linearly independent?
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\itemitem{b)} Is $\{v_1, v_2, v_3\}$ linearly independent?
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\itemitem{c)} Let $u = (0,1)$.  Express $u$ as a linear combination of $v_1, v_2$.
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2.\ \ Let $\Bbb V$ be the vector space of all $2 \times 2$ matrices and $\Bbb W$ be the
subset of $\Bbb V$ such that 
$$\Bbb W = \{A \in \Bbb V| A_{11} + A_{22} = 0\}.$$
\itemitem{a)}Show that $\Bbb W$ is a vector space.
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\itemitem{b)} What is the dimension of $\Bbb W$?
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\itemitem{c)}Find a basis for $\Bbb W$.

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3.\ \ Let $\Cal P_2$ be the vector space of all polynomials of degrees less or equal to
2.  Let $f_0(x) = 1,\ f_1(x) = x+1.\ \   f_2(x) = x^2+x+1$
\itemitem{a)} Prove that $\Cal B= \{f_0, f_1, f_2\}$ is a basis for $\Cal P_2$.
\vfill
\itemitem{b)}Let $f(x) = x^2-x$.  What are the coordinates of $f$ in the ordered
basis $\Cal B$?

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4.\ \ Let $T : R^4 \longrightarrow R^4$ be a linear transformatin defined by $T \Bbb
X = A \Bbb X$ where

$$A = \left[ \matrix 1&0&2&1\\
1&1&3&1\\
0&1&1&0\\
0&2&2&3 \endmatrix
\right]$$
\itemitem{a)}Find the Null space of $T, N_T$.
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\itemitem{b)}Find a basis for $N_T$.
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\itemitem{c)}Find a basis for $R_T$, the range of $T$.
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5.\ \ Let $T$ be a linear operator on $R^2$ such that $T(e_1) = (1,2), T(e_2) =
(2,1)$.
\itemitem{a)}Find $T(x,y)$ for any $(x,y) \in R^2$.
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\itemitem{b)}Is $T$ invertible?  If it is, find $T^{-1}$. 
                                                                                                                                               
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