Find an orthogonal basis for ℝ3 that begins with an orthogonal basis for Col()

Question

The problem is as follows:

Find an orthogonal basis for ℝ3 that begins with an orthogonal basis for Col(A). The matrix A is given in the image below (do not know how to display matrices correctly in text).

My calculations

Is this a correct answer to the problem? Or do i need to use the Gram-Schmidt process directly on Col(A) before i find the reduced row echelon form of A?

Answer 1

The span of the two vectors you found is $$ \operatorname{span}\left\{\begin{bmatrix} 1\\0\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\} =\left\{\begin{bmatrix}x\\ y\\0\end{bmatrix}:\ x,y\in\mathbb R\right\}. $$ So your candidate for the basis cannot span neither the second nor the third column.

You have to do Gram-Schmidt, not forgetting that you will often (as in this case) get some zeros; since the dimension of col$(A)$ is $2$.