I have a Linear homework questions asking what the QR factorization of a matrix A whose columns are orthogonal but not of unit norm might look like. I reread the section in textbook about norms, but still don't get how vectors cannot be of unit norm (I'm assuming this means 1-norm). Doesn't this just mean that I'm adding the absolute values of all values in each vector? So how could a vector not have it's values added? In my mind every vector is of unit norm, no?
Thanks in advance, I'm sure the question is silly but I had to ask.
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$\begingroup$As has been stated in the comments, a vector of unit norm has length one: $||v||_1=1$. The subscribt one here denotes we're talking about the 1-norm as you are suggesting. So if $$v=\begin{pmatrix}3\\4\\-5\end{pmatrix},$$ then the 1-norm of this vector will be the sum of its elements in absolute value, so: $$|3|+|4|+|-5|=12.$$
Since 12 is not equal to 1, this vector is not of unit norm.
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