Can someone give me the right hint to understand understand this proof?
To be precise, I don't understand why "Since any vector $v$ can be expressed as a linear combination of the singular vectors plus a vector perpendicular to the $v_i$".
2026-03-25 07:38:10.1774424290
SVD explaination
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In the cited passage, $A$ is an $n\times d$ matrix of rank $r$, with $r$ non-zero singular values. The $r$ vectors $v_i$ might not span the full $d$-dimensional space $\mathbb R^d $. For instance, if $r<d$. The author is evidently covering that possibility.