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 $T(\mathbf{x})$.

 
LoadingAdd to solve later

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
\[T(\mathbf{x})=\begin{bmatrix}
x_1+2x_2+3x_3 \\
4x_1+5x_2+6x_3
\end{bmatrix}.\]


LoadingAdd to solve later

Sponsored Links

More from my site

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Problems and solutions in Linear Algebra
Linear Properties of Matrix Multiplication and the Null Space of a Matrix

Let $A$ be an $m \times n$ matrix. Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in...

Close