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

Ohio State University exam problems and solutions in mathematics

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


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  1. 04/06/2017

    […] Problem 5 and its solution: Determine value of linear transformation from $R^3$ to $R^2$ […]

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