## Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$

## Problem 605

Let $T:\R^2 \to \R^3$ be a linear transformation such that

\[T\left(\, \begin{bmatrix}

3 \\

2

\end{bmatrix} \,\right)

=\begin{bmatrix}

1 \\

2 \\

3

\end{bmatrix} \text{ and }

T\left(\, \begin{bmatrix}

4\\

3

\end{bmatrix} \,\right)

=\begin{bmatrix}

0 \\

-5 \\

1

\end{bmatrix}.\]

**(a)** Find the matrix representation of $T$ (with respect to the standard basis for $\R^2$).

**(b)** Determine the rank and nullity of $T$.

*(The Ohio State University, Linear Algebra Midterm)*

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