Linear Algebra

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

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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