Projection and orthogonal residual

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Problem

Let $u=(1,1,0)^{\mathsf T}$ and $v=(2,0,1)^{\mathsf T}$ . Compute the orthogonal projection of $v$ onto $\operatorname{span}(u)$ and the residual. Verify the residual is orthogonal to $u$ .

Reveal answer or reference solution

\operatorname{proj}_u(v)=\frac{v^{\mathsf T}u}{u^{\mathsf T}u}u =\frac2{2}u=(1,1,0)^{\mathsf T}

Hence the residual is

r=v-\operatorname{proj}_u(v)=(1,-1,1)^{\mathsf T}

The check is $u^{\mathsf T}r=1-1+0=0$ , so the residual is orthogonal to the projection subspace.

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Projection and orthogonal residual

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