Matrix Representation of a Linear Transformation of the Vector Space $R^2$ to $R^2$

Linear algebra problems and solutions

Problem 255

Let $B=\{\mathbf{v}_1, \mathbf{v}_2 \}$ be a basis for the vector space $\R^2$, and let $T:\R^2 \to \R^2$ be a linear transformation such that
\[T(\mathbf{v}_1)=\begin{bmatrix}
1 \\
-2
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
3 \\
1
\end{bmatrix}.\]

If $\mathbf{e}_1=\mathbf{v}_1+2\mathbf{v}_2 \text{ and } \mathbf{e}_2=2\mathbf{v}_1-\mathbf{u}_2$, where $\mathbf{e}_1, \mathbf{e}_2$ are the standard unit vectors in $\R^2$, then find the matrix of $T$ with respect to the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$.
 
LoadingAdd to solve later

Solution.

The matrix representation $A$ of the linear transformation $T$ with respect to the basis $\{\mathbf{e}_1, \mathbf{e}_2 \}$ is given by
\[A=\begin{bmatrix}
T(\mathbf{e}_1) & T(\mathbf{e}_2)
\end{bmatrix}. \tag{*}\]

To find the vectors $T(\mathbf{e}_1), T(\mathbf{e}_2)$ we use the linearity of $T$.
We have
\begin{align*}
T(\mathbf{e}_1)&=T(\mathbf{v}_1+2\mathbf{v}_2)\\
&=T(\mathbf{v}_1)+2T(\mathbf{v}_2) \qquad \text{ (by the linearity of $T$)}\\
&=\begin{bmatrix}
1 \\
-2
\end{bmatrix} +2\begin{bmatrix}
3 \\
1
\end{bmatrix}\\
&=\begin{bmatrix}
7 \\
0
\end{bmatrix}.
\end{align*}


Also we have
\begin{align*}
T(\mathbf{e}_2)&=T(2\mathbf{v}_1-\mathbf{u}_2)\\
&=2T(\mathbf{v}_1)-T(\mathbf{u}_2)\\
&=2\begin{bmatrix}
1 \\
-2
\end{bmatrix}-\begin{bmatrix}
3 \\
1
\end{bmatrix}\\
&=\begin{bmatrix}
-1 \\
-5
\end{bmatrix}.
\end{align*}


Therefore the matrix $A=\begin{bmatrix}
T(\mathbf{e}_1) & T(\mathbf{e}_2)
\end{bmatrix}$ of the linear transformation $T$ is given by
\[A=\begin{bmatrix}
7 & -1\\
0& -5
\end{bmatrix}.\]


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Your email address will not be published.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
Linear Algebra Problems and Solutions
Find the Distance Between Two Vectors if the Lengths and the Dot Product are Given

Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\R^n$ such that their length are \[\|\mathbf{a}\|=\|\mathbf{b}\|=1\] and the inner product \[\mathbf{a}\cdot \mathbf{b}=\mathbf{a}^{\trans}\mathbf{b}=-\frac{1}{2}.\]...

Close