I'm having issues with understanding one of the exercises I'm making.
I have to find an orthogonal basis for the column space of $A$, where:
$A = \begin{bmatrix} 0 & 2 & 3 & -4 & 1\\ 0 & 0 & 2 & 3 & 4 \\ 2 & 2 & -5 & 2 & 4\\ 2 & 0 & -6 & 9 & 7 \end{bmatrix}.$
The first question was to find a basis of the column space of $A$, clearly this is simply the first $3$ column vectors (by reducing it to row echelon form, and finding the leading $1$'s).
However, then I had to find an orthogonal basis out of the column space of $A$, and here is where I get lost. I started off with finding the first vector:
$u_1 = \begin{bmatrix}0\\0\\2\\2\\\end{bmatrix}.$
Then I thought I would find the second vector like this:
$u_2 = \begin{bmatrix}2\\0\\2\\0\\\end{bmatrix}-\left(\begin{bmatrix}2\\0\\2\\0\\\end{bmatrix}\cdot\begin{bmatrix}0\\0\\2\\2\\\end{bmatrix}\right)*\begin{bmatrix}0\\0\\2\\2\\\end{bmatrix} = \begin{bmatrix}2\\0\\2\\0\\\end{bmatrix}-4*\begin{bmatrix}0\\0\\2\\2\\\end{bmatrix} = \begin{bmatrix}2\\0\\-6\\-8\\\end{bmatrix}.$
However, according to the result sheet we were given, instead of having a $4$, I should have $\frac{4}{8}$. I somehow can not figure out what I am missing, since the dot product of the two vectors clearly is $4$.
Also, as a second question: if I had to find a orthonormal basis I would only have to take the orthogonal vectors found here, and multiply them by their $1$/length, correct?