3.4.3

Consider the standard basis $$\textbf{E} = (e_1,e_2) $$ of $$\mathbb{F}^2 $$, and the new basis $$\textbf{E'} = (e_1 + e_2, e_1 - e_2) $$. Observing that $$\frac{1}{2} (e_1 + e_2) + \frac{1}{2} (e_1 - e_2) = e_1 $$, and $$\frac{1}{2}(e_1 + e_2) - \frac{1}{2}(e_1 - e_2) = e_2 $$, we can then write:

$$(e_1 + e_2, e_1 - e_2) \begin{bmatrix}

\frac{1}{2} & \frac{1}{2} \\

\frac{1}{2} & -\frac{1}{2} \\

\end{bmatrix} = (e_1,e_2) $$.

Thus the change of basis matrix P when going from the basis E to the basis E' is the matrix in the equation above.