3.7: Orthogonal Projections
- Page ID
- 96157
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)View Orthogonal Projections on YouTube
Suppose that \(V\) is an \(n\)-dimensional vector space and \(W\) is a \(p\)-dimensional subspace of \(V\). Let \(\{\text{s}_1,\: \text{s}_2,\:\cdots , \text{s}_p\}\) be an orthonormal basis for \(W\). Extending the basis for \(W\), let \(\{\text{s}_1,\: \text{s}_2,\cdots , \text{s}_p,\: \text{t}_1,\: \text{t}_2,\cdots , \text{t}_{n−p}\}\) be an orthonormal basis for \(V\).
Any vector \(\text{v}\) in \(V\) can be written in terms of the basis for \(V\) as
\[\text{v}=a_1\text{s}_1+a_2\text{s}_2+\cdots +a_p\text{s}_p+b_1\text{t}_1+b_2\text{t}_2+b_{n-p}\text{t}_{n-p}.\nonumber \]
The orthogonal projection of \(\text{v}\) onto \(W\) is then defined as
\[\text{v}_{\text{proj}_w}=a_1\text{s}_1+a_2\text{s}_2+\cdots +a_p\text{s}_p,\nonumber \]
that is, the part of \(\text{v}\) that lies in \(W\).
If you only know the vector \(v\) and the orthonormal basis for \(W\), then the orthogonal projection of \(\text{v}\) onto \(W\) can be computed from
\[\text{v}_{\text{proj}_w}=(\text{v}^{\text{T}}\text{s}_1)\text{s}_1+(\text{v}^{\text{T}}\text{s}_2)\text{s}_2+\cdots +(\text{v}^{\text{T}}\text{s}_p)\text{s}_p,\nonumber \]
that is, \(a_1=\text{v}^{\text{T}}\text{s}_1,\: a_2=\text{v}^{\text{T}}\text{s}_2\), etc.
We can prove that the vector \(\text{v}_{\text{proj}_W}\) is the vector in \(W\) that is closest to \(\text{v}\). Let \(\text{w}\) be any vector in \(W\) different than \(\text{v}_{\text{proj}_W}\), and expand \(\text{w}\) in terms of the basis vectors for \(W\):
\[\text{w}=c_1\text{s}_1+c_2\text{s}_2+\cdots +c_p\text{s}_p.\nonumber \]
The distance between \(\text{v}\) and \(\text{w}\) is given by the norm \(||\text{v} − \text{w}||\), and we have
\[\begin{aligned}||\text{v}-\text{w}||^2&=(a_1-c_1)^2+(a_2-c_2)^2+\cdots +(a_p-c_p)^2+b_1^2+b_2^2+\cdots +b_{n-p}^2 \\ &\geq b_1^2+b_2^2+\cdots +b_{n-p}^2=||\text{v}-\text{v}_{\text{proj}_w}||^2,\end{aligned} \nonumber \]
or \(||\text{v}-\text{v}_{\text{proj}_w}||\leq ||\text{v}-\text{w}||\), a result that will be used later in the problem of least squares.