Give a Formula For a Linear Transformation From $\R^2$ to $\R^3$
Problem 339
Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a basis of the vector space $\R^2$, where
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
1
\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}
1 \\
-1
\end{bmatrix}.\]
The action of a linear transformation $T:\R^2\to \R^3$ on the basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ is given by
\begin{align*}
T(\mathbf{v}_1)=\begin{bmatrix}
2 \\
4 \\
6
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
0 \\
8 \\
10
\end{bmatrix}.
\end{align*}
Find the formula of $T(\mathbf{x})$, where
\[\mathbf{x}=\begin{bmatrix}
x \\
y
\end{bmatrix}\in \R^2.\]
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