The book said to use equation 6, but I don’t know how it applies. When I tried to apply equation 6, I kept getting the wrong answer. Could you please walk me through this problem? Thanks!

Use the fact that:

\[\begin{bmatrix}

1 \\

0

\end{bmatrix}=\frac{1}{2}\left(\,\begin{bmatrix}

1 \\

1

\end{bmatrix}+\begin{bmatrix}

1 \\

-1

\end{bmatrix}\,\right)\]
\[\begin{bmatrix}

0 \\

1

\end{bmatrix}=\frac{1}{2}\left(\,\begin{bmatrix}

1 \\

1

\end{bmatrix}-\begin{bmatrix}

1 \\

-1

\end{bmatrix}\,\right).\]
Find $T\left(\, \begin{bmatrix}

1 \\

0

\end{bmatrix} \,\right)$ and $T\left(\, \begin{bmatrix}

0 \\

1

\end{bmatrix} \,\right)$ and write $T\left(\, \begin{bmatrix}

x_1 \\

x_2

\end{bmatrix} \,\right)$ as a linear combination of these two.

Hope that helps!

Different methods to solve this kind of problems are explained in Find a general formula of a linear transformation from $\R^2$ to $\R^3$

Thank you! That makes sense!