Linear transformation and matrix

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

1. Yes
2. No

Ans: A

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

1. $\bigl(\vec{a}\quad \vec{b}\bigr)$

2. $\bigl(\begin{smallmatrix} \vec{a}^t \\ \vec{b}^t \end{smallmatrix} \bigr)$

3. None of the above.

Ans: A