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Q6. Let W be the subspace of R' spanned by the vectors u. = 3(1, -1,1,1),...




Q6. Let W be the subspace of R spanned by the vectors u. = 3(1, -1,1,1), uz = 5(–1,1,1,1). (a) Check that {uj,uz) is an orth
Q6. Let W be the subspace of R' spanned by the vectors u. = 3(1, -1,1,1), uz = 5(–1,1,1,1). (a) Check that {uj,uz) is an orthonormal set using the dot product on R. (Hence it forms an orthonormal basis for W.) (b) Let w = (-1,1,5,5) EW. Using the formula in the box above, express was a linear combination of u and u. (c) Let v = (-1,1,3,5) = R'. Find the orthogonal projection of v onto W.
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(۱٫۱ را را-) = 4 (ارا را- ۱) { = ا :: )+(!)*(4)(!) Mu/۱ - ): ): ):( | = ال۱۱ = 11,ا|1 (54) ;({}() :()({ ;({}( دهار، با = با =© To find the orthogonal projection, we first find the arthogonal complement of W. lut cu,y,z,w) E WF. then <cury, z, w) C1,-- ا ا ا \\ - = ر- +%2 = 6 Solving these two we get ( 23 ) =5] This projectionſ is آ = (5ر3 رار ا 5 + (1 را ر-ر1) 3 . 1 (۱٫۱ ر

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