= 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?

  A. Yes
  A. 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

  A. $\left(\vec{a}\quad \vec{b}\right)$
  A. $\left(\begin{smallmatrix} \; \vec{a}^t \; \\ \; \vec{b}^t \; \end{smallmatrix} \right)$
  A. None of the above.

Ans: A