Let $\alpha = (x_{1},x_{2})$ and $\beta = (y_{1},y_{2})$ be vectors in $\textbf{R}^{2}$ such that
\begin{align*}
(x_{1}y_{1} + x_{2}y_{2} = 0)\wedge (x^{2}_{1} + x^{2}_{2} = y^{2}_{1} + y^{2}_{2} = 1)
\end{align*}
Prove that $\mathcal{B} = \{\alpha,\beta\}$ is a basis for $\textbf{R}^{2}$. Find the coordinates of the vector $(a,b)$ in the ordered basis $\mathcal{B}$.
MY ATTEMPT
Since $x^{2}_{1} + x^{2}_{2} = 1$, we can consider that $x_{1} = \cos(\theta)$ and $x_{2} = \sin(\theta)$. Analogously, $y_{1} = \cos(\varphi)$ and $y_{2} = \sin(\varphi)$. Moreover, according to the the first condition, one has
\begin{align*}
\cos(\theta)\cos(\varphi) + \sin(\theta)\sin(\varphi) = 0 \Longleftrightarrow \cos(\theta – \varphi) = 0 \Longleftrightarrow \theta = \varphi + \frac{\pi}{2}
\end{align*}
Having said that, we have that $\alpha = (-\sin(\varphi),\cos(\varphi))$ and $\beta = (\cos(\varphi),\sin(\varphi))$. Thus it suffices to prove that $\alpha$ and $\beta$ are linearly independent, because $\textbf{R}^{2}$ has dimension two. Indeed, this is the case
\begin{align*}
& a\alpha + b\beta = (-a\sin(\varphi) + b\cos(\varphi),a\cos(\varphi) + b\sin(\varphi)) = (0,0) \Longrightarrow\\\\
& [(a^{2}+b^{2})\cos^{2}(\varphi) = 0)] \wedge [(a^{2}+b^{2})\sin^{2}(\varphi) = 0] \Longrightarrow a = b = 0
\end{align*}
Hence $\mathcal{B}$ is a basis for $\textbf{R}^{2}$. Consequently, the following relation holds
\begin{align*}
\begin{bmatrix}
a'\\
b'
\end{bmatrix} =
\begin{bmatrix}
-\sin(\varphi) & \cos(\varphi)\\
\cos(\varphi) & \sin(\varphi)
\end{bmatrix}^{-1}
\begin{bmatrix}
a\\
b
\end{bmatrix} =
\begin{bmatrix}
-\sin(\varphi) & \cos(\varphi)\\
\cos(\varphi) & \sin(\varphi)
\end{bmatrix}
\begin{bmatrix}
a\\
b
\end{bmatrix}
\end{align*}
Am I reasoning correctly?
Best Answer
My take: $\langle \alpha,\beta\rangle=0$ (and $\alpha, \beta\ne0$) $\implies \alpha$ and $\beta$ are linearly independent, since they're not parallel. Hence they form a basis.
Given $(a,b)$, use the change of basis matrix to get the coordinates. The change of basis matrix is the one whose columns are $\alpha$ and $\beta$ respectively (written in the standard basis).