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Let $$v _1 = (1, 1), v _2 = (1, 3)$$ Let $x$ and $y$ be the coordinates with respect to the standard basis: $(0,1)$ and $(1,0)$ and let $u$ and $v$ be the coordinates with respect to $v_1, v_2$. Write the equations to translate from $(x, y)$ to $(u, v)$ and back. I have the equations by solving the system

\begin{array}{rcl} u+v=x \\ u+3v=y\end{array}

Salech Alhasov
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user278039
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1 Answers1

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From your system you have: $$ \begin{cases} u=\frac{3}{2}x-\frac{1}{2}y\\ v=-\frac{1}{2}x+\frac{1}{2}y \end{cases} $$ so the matrices that represent the change of basis are: $$ M=\begin{bmatrix} 1&1\\ 1&3 \end{bmatrix} \qquad M^{-1}=\begin{bmatrix} \frac{3}{2}&-\frac{1}{2}\\ -\frac{1}{2}&\frac{1}{2} \end{bmatrix} $$

Emilio Novati
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