Linear transformation and matrix

A matrix represents a linear transformation of vectors. Is the reverse true? Namely, can any linear transformation (of a finite dimensional vectors) be represented by a matrix?

Matrix and linear transformation

Suppose that a linear transformation of a 2D vector space maps $\bigl(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix} \bigr)$ to $\vec{a}$, and $\bigl(\begin{smallmatrix} 0 \\ 1 \end{smallmatrix} \bigr)$ to $\vec{b}$. (Recall that we consider $\vec{a}$ and $\vec{b}$ as column vectors.) The matrix that represents this linear transformation is

$\bigl(\vec{a}\quad \vec{b}\bigr)$
$\bigl(\begin{smallmatrix} \vec{a}^t \\ \vec{b}^t \end{smallmatrix} \bigr)$
None of the above.

-  ⇤ ← Revision 7 as of 9:29AM, Sep 27, 2011 → 
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+   ← Revision 8 as of 9:30AM, Sep 27, 2011 → ⇥
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-Suppose that a linear transformation of a 2D vector space maps $\bigl(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix} \bigr)$ to $\vec{a}$, and $\bigl(\begin{smallmatrix} 0 \\ 1 \end{smallmatrix} \bigr)$ to $\vec{b}$.  The matrix that represents this linear transformation is
+Suppose that a linear transformation of a 2D vector space maps $\bigl(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix} \bigr)$ to $\vec{a}$, and $\bigl(\begin{smallmatrix} 0 \\ 1 \end{smallmatrix} \bigr)$ to $\vec{b}$.  (Recall that we consider $\vec{a}$ and $\vec{b}$ as column vectors.)  The matrix that represents this linear transformation is