[Math] Orthonormal and/or Orthogonal Basis of a Pair of Vectors

Question

I was hoping someone could verify if this is the correct way to answer this problem:

Let $\mathbb{R^{2}}$ have the standard dot product. Classify the following pair of vectors as (i) basis, (ii) orthogonal basis and/or, (iii) orthonormal basis:

$$
\vec{v_{1}} = \begin{bmatrix} -1 \\ 2 \end{bmatrix} \quad \vec{v_{2}} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}
$$

The way I did it was:

$$
\textbf{Basis:} \qquad \text{rref}\left( \begin{bmatrix} -1 & 2 \\ 2 & 1 \end{bmatrix} \right) \,\,=\,\, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \quad \checkmark \\
\textbf{Orthogonal Basis:} \qquad \vec{v_{1}} \cdot \vec{v_{2}} \,\,=\,\, \begin{bmatrix} -1 \\ 2 \end{bmatrix} \cdot \begin{bmatrix} 2 \\ 1 \end{bmatrix} \,\,=\,\, 0 \quad \checkmark \\
\textbf{Orthonormal Basis:} \qquad \Vert\vec{v_{1}}\Vert \,\,=\,\,\sqrt{5} \qquad \Vert\vec{v_{2}}\Vert \,\,=\,\, \sqrt{5} \qquad \textbf{X}
$$

so this pair of vectors is classified as an $\boxed{\text{orthogonal basis}}$

I know that the final answer is correct from the back of the book that I got this problem from, but if I didn't get the solution the right way I would like to know where I went wrong.

Thank you in advanced

Answer 1

Your answer is correct. Just note that in order for a set to form a basis, it must be linearly independent, and span the given space. It is easy to see in this example, but in further problems, you may need to verify manually.