Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$

Linear Transformation problems and solutions

Problem 156

Let $T: \R^3 \to \R^2$ be a linear transformation such that
\[T(\mathbf{e}_1)=\begin{bmatrix}
1 \\
4
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
2 \\
5
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
3 \\
6
\end{bmatrix},\] where
\[\mathbf{e}_1=\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}, \mathbf{e}_3=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}\] are the standard unit basis vectors of $\R^3$.
For any vector $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}\in \R^3$, find a formula for $T(\mathbf{x})$.

 
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Hint.

Recall the definition of a linear transformation $T: \R^3 \to \R^2$. A map $T$ is a linear transformation if the map $T$ satisfies:

  1. $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$ for any $\mathbf{u}, \mathbf{v}\in \R^3$, and
  2. $T(c\mathbf{v})=cT(\mathbf{v})$ for any $\mathbf{v} \in \R^3$ and $c\in \R$.

Solution.

Using the standard unit basis vectors, any vector $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}\in \R^3$ can be expressed as a linear combination
\[\mathbf{x}=x_1\mathbf{e}_1+x_2\mathbf{e}_2+x_3\mathbf{e}_3.\] Since $T$ is a linear transformation, we have
\begin{align*}
T(\mathbf{x})&=T(x_1\mathbf{e}_1+x_2\mathbf{e}_2+x_3\mathbf{e}_3)\\
&=x_1T(\mathbf{e}_1)+x_2T(\mathbf{e}_2)+x_3T(\mathbf{e}_3)\\[6pt] &=x_1\begin{bmatrix}
1 \\
4
\end{bmatrix}+x_2\begin{bmatrix}
2 \\
5
\end{bmatrix}+x_3\begin{bmatrix}
3 \\
6
\end{bmatrix}\\[6pt] &=\begin{bmatrix}
x_1+2x_2+3x_3 \\
4x_1+5x_2+6x_3
\end{bmatrix}.
\end{align*}
Therefore the formula is given by
\[T(\mathbf{x})=\begin{bmatrix}
x_1+2x_2+3x_3 \\
4x_1+5x_2+6x_3
\end{bmatrix}.\]


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