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A dot product is given in the space $\mathbb{R}^2$ with the following formula:

$$\left\langle\begin{bmatrix}x_1\\x_2\end{bmatrix}, \begin{bmatrix}y_1\\y_2\end{bmatrix}\right\rangle = 2x_1y_1 + x_2y_2$$

Is the matrix $\begin{bmatrix}1 & 1\\2 & 1\end{bmatrix}$ in this dot product orthogonal?

I'm not sure if I know what this exercise is asking from me. I know we can prove that a matrix is orthogonal if: $AA^T=I$ holds or by showing that each vector in the matrix is a unit vector (it's length is 1) and the dot product of each pair of vectors in the matrix is zero, but I think this isn't the case here.

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Let the columns of the matrix $\mathbf A$ be $\mathbf a_1$ and $\mathbf a_2$. Consider what the condition $\mathbf A^T\mathbf A=\mathbf I$ is really telling you: $$\mathbf A^T\mathbf A=\begin{bmatrix}\mathbf a_1&\mathbf a_2\end{bmatrix}^T\begin{bmatrix}\mathbf a_1&\mathbf a_2\end{bmatrix}=\begin{bmatrix}\mathbf a_1^T\\\mathbf a_2^T\end{bmatrix}\begin{bmatrix}\mathbf a_1&\mathbf a_2\end{bmatrix}=\begin{bmatrix}\mathbf a_1^T\mathbf a_1&\mathbf a_1^T\mathbf a_2\\\mathbf a_2^T\mathbf a_1&\mathbf a_2^T\mathbf a_2\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}.$$ Compare the entries of the last two matrices: $\mathbf a_1^T\mathbf a_1=1$, $\mathbf a_1^T\mathbf a_2=0$, $\mathbf a_2^T\mathbf a_2=1$. This means precisely that the columns of the matrix are orthonormal with respect to the usual inner product $\langle\mathbf x,\mathbf y\rangle=\mathbf x^T\mathbf y=x_1y_1+x_2y_2$. It does not tell you anything about whether they are orthonormal with respect to the different inner product $\langle\mathbf x,\mathbf y\rangle=2x_1y_1+x_2y_2$ that you've been asked to use! So what you need to do is check that orthonormality holds with respect to this inner product instead: $\langle\mathbf a_1,\mathbf a_1\rangle=1$, $\langle\mathbf a_1,\mathbf a_2\rangle=0$, $\langle\mathbf a_1,\mathbf a_2\rangle=1$.

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