jhenezhubby01ff

2022-09-09

If the subspace is described as the range of a matrix: $S=\{Ax:x\in {\mathbb{R}}^{n}\}$, then the orthogonal complement is the set of vectors orthogonal to the rows of $$A$$, which is the nullspace of ${A}^{T}$. How to make the above claim from the definition of orthogonal complement as the set of vectors that are orthogonal to all $Ax$.

Terahertztl

Beginner2022-09-10Added 8 answers

Let ${C}_{1},\dots ,{C}_{p}$ be the columns of the matrix $A$, then $S=\mathrm{span}({C}_{1},\dots ,{C}_{p})$, hence $x\in {S}^{\perp}$ iff $\u27e8{C}_{i},x\u27e9=0,\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall}i\in \{1,\dots ,p\}$ iff ${A}^{T}x=0$. (the lines of ${A}^{T}$ are the columns of $A$).

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