Orthogonal bases• Recall: Suppose that v1,, vnare nonzero and (pairwise) orthogonal . Then v1,,vnare independent.Definition 1. A basis v1,, vnof a vector space V is an orthogonal basis if thevectors are (pairwise) orthogonal.Example 2. Are t he vectors1−10,110,001an orthogonal basis for R3?Solution.Note that we do not need to check that the three vectors are independent. That followsfrom their orthogonality.Example 3. Suppos e v1,, vnis an orthogonal ba sis of V , and th a t w is in V . Findc1,, cnsuch thatw = c1v1++ cnvn.Solution. Take the dot product of v1with both sides:If v1,, vnis an orthogonal ba sis of V , and w is in V , th e nw = c1v1++ cnvnwith cj=w · vjvj· vj.Armin [email protected] 4. Express374in terms of the basis1−10,110,001.Solution.Definition 5. A basis v1,, vnof a vector space V is a n orthonormal basis if thevectors are orthogonal and have length1.If v1,, vnis an orthonormal b asis of V , and w is in V , t henw = c1v1++ cnvnwith cj= vj· w.Example 6. Is the basis1−10,110,001orthonormal? If not, normalize the vectorsto produce an orthonormal basis .Solution.Armin [email protected] thogon al projecti onsyxDefinition 7. The orthogonal projection of vector x ontovector y isxˆ=x · yy · yy.• The vector xˆis the c losest v ec tor to x, which is inspan{y }.• The “error” x⊥= x − xˆis orthogonal to span{y }.x⊥= x − xˆ= x −x · yy · yy is also referred to as the component of xorthogonal to y.Example 8. What is the orthogonal projection of x =−84onto y =31?Solution.Example 9.What are the orthogonal projections of211onto each of the vectors1−10,110,001?Solution.Armin [email protected]: If v1,, vnis an orthogonal ba sis of V , and w is in V , th e nw = c1v1++ cnvnwith cj=w · vjvj· vj. wdecomposes as the sum of its projections ont o each basis vectorOr thogon al projecti on on subspacesTheorem 10. Let W be a subspace of Rn. Then, each x in Rncan be uniquely writtenasx = xˆin W+ x⊥in W⊥.v1v2xˆxx⊥• xˆis the orthogonal projection of x onto W .• xˆis the point in W closest to x. For any other y in W ,dist(x, xˆ) < dist(x, y).• If v1,, vmis an orthogonal basis of W , thenExample 11. Let W = span(301,010), and x =0310.• Find the orthogonal projection of x onto W .• Wr ite x as a vector in W plus a ve c tor orthogonal to W .Armin [email protected] 12. Let v1,, vmbe an orthogonal basis of W , a subspace of Rn. Theprojection m a p πW: Rn→ Rn, given byxx · v1v1· v1v1++x · vmvm· vmvmis linear. The matr ix P representing πWwith re spect to t he sta ndard basis is th ecor r e sponding project ion matrix.Example 13. Find the projection matrix P which correspon ds to orthogonal projectionontoW = span(301,010)in R3.Solution.Armin [email protected] 14. Compute P2for the projection matrix we just computed. Exp l ain!Solution.Practice problemsExample 15. Find the closest point to x in span{v1, v2}, wherex =240−2, v1=1100, v2=0011.Solution. This is the orthogonal projection of x onto span{v1, v2}.Armin
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