# Determine a Value of Linear Transformation From $\R^3$ to $\R^2$

## Problem 368

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

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

0 \\

1 \\

0

\end{bmatrix}\,\right) =\begin{bmatrix}

1 \\

2

\end{bmatrix} \text{ and }T\left(\, \begin{bmatrix}

0 \\

1 \\

1

\end{bmatrix}\,\right)=\begin{bmatrix}

0 \\

1

\end{bmatrix}. \]
Then find $T\left(\, \begin{bmatrix}

0 \\

1 \\

2

\end{bmatrix} \,\right)$.

(*The Ohio State University, Linear Algebra Exam Problem*)

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## Solution.

We first express the vector $\begin{bmatrix}

0 \\

1 \\

2

\end{bmatrix}$ as a linear combination

\[\begin{bmatrix}

0 \\

1 \\

2

\end{bmatrix}=c_1\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}+c_2\begin{bmatrix}

0 \\

1 \\

1

\end{bmatrix}.\]
Then we find that $c_1=-1$ and $c_2=2$. Hence we obtain

\[\begin{bmatrix}

0 \\

1 \\

2

\end{bmatrix}=-\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}+2\begin{bmatrix}

0 \\

1 \\

1

\end{bmatrix}.\]

We now compute

\begin{align*}

T\left(\, \begin{bmatrix}

0 \\

1 \\

2

\end{bmatrix} \,\right)

&=T\left(\, -\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}+2\begin{bmatrix}

0 \\

1 \\

1

\end{bmatrix} \,\right)\\

&=-T\left(\, \begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix} \,\right)+2\left(\, \begin{bmatrix}

0 \\

1 \\

1

\end{bmatrix} \,\right) && \text{by linearity of $T$}\\

&=-\begin{bmatrix}

1 \\

2

\end{bmatrix}+2\begin{bmatrix}

0 \\

1

\end{bmatrix}\\

&=\begin{bmatrix}

-1 \\

0

\end{bmatrix}.

\end{align*}

Therefore we have found that

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

0 \\

1 \\

2

\end{bmatrix} \,\right)=\begin{bmatrix}

-1 \\

0

\end{bmatrix}\]

## Linear Algebra Midterm Exam 2 Problems and Solutions

- True of False Problems and Solutions: True or False problems of vector spaces and linear transformations
- Problem 1 and its solution: See (7) in the post “10 examples of subsets that are not subspaces of vector spaces”
- Problem 2 and its solution: Determine whether trigonometry functions $\sin^2(x), \cos^2(x), 1$ are linearly independent or dependent
- Problem 3 and its solution: Orthonormal basis of null space and row space
- Problem 4 and its solution: Basis of span in vector space of polynomials of degree 2 or less
- Problem 5 and its solution (current problem): Determine value of linear transformation from $R^3$ to $R^2$
- Problem 6 and its solution: Rank and nullity of linear transformation from $R^3$ to $R^2$
- Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$
- Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space

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