MATH115 Lecture 29: lect115_29_rev_f14
10 Oct 2015
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Monday, november 10 lecture 29 : orthogonal complements and projections. Note: in the notes below < w, x > means w dot product x . 29. 1 definition a vector z in n which is orthogonal to every vector in a subspace w of. N is called an orthogonal complement to w. the set of all orthogonal complements of. That is, w = {x in n : < w, x > = 0 for all w in w}, pronounced. 29. 2 proposition suppose w = span{v1, v2, , vn} in m. Then x w if and only if. < vi, x > = 0 for all i = 1 to n. By definition of w , if x w then < v, x > = 0 for all v in w hence < vi, x > = 0 for all i = 1 to n. Then v = x1v1 + x2 v1 + + xnvn (since w = span{v1, v2, , vn})
