In matlab, we are asked to set A=rand(5,2)*rand(2,5)
then to set Q=orth(A), w=null(A'), S=[Q W]
the matrix S should be orthogonal. Why? (I have no clue on how to answer this)
2026-03-29 05:10:35.1774761035
Fundamental Subspaces: Orthonormal Bases
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$A$ is an arbitrary matrix. We note that $$ \text{null}(A^*) = \text{image}(A)^\perp $$ So, the columns of $Q$ form an orthonormal basis of $\text{image}(A)$, and the columns of $W$ form an orthonormal basis of $\text{image}(A)^\perp$. Together, the columns of $[Q\;\ W]$ are an orthonormal basis of $\Bbb R^n$, which means that the matrix is orthogonal.