Inner productsOctober 8, 2007Inner products are used to deal with two important geometric notions:angle and distance.Definition 1 Let v and w be vectors in Rn, with coordinates (x1, . . . xn)and (y1, . . . yn), respectively. Then the inner product or dot product of v andw is:v · w := hv, wi := (v|w) := x1y1+ x2y2+ · · · xnyn.Note that there are at least three sets of notation commonly used to denotethe inner product. Furthermore, it is sometimes called the “dot product” or“scalar product.” Note that the dot product of two vectors in Rnis a realnumber, and that the matrix product vTw is the 1 × 1 matrix whose onlyentry is (v|w). We shall sometimes identify the matrix vTw with the number(v|w).Theorem 2 The inner produc t Rn× Rn→ R satisfies the following prop-erties:1. (v + v0|w) = (v|w) + (v0|w) if v, v0, w ∈ Rn.2. (av|w) = a(v|w) if v, w ∈ Rnand a ∈ R.3. (v|w) = (w|v) if v, w ∈ Rn.4. (v|v) > 0 if v 6= 0, for any v ∈ Rnand (0|0) = 0.These properties are quite easy to verify. Note however that the last of themuses the special fact that the square of any real number is positive.1Definition 3 If v ∈ Rn, then ||v|| :=q(v|v). The (nonnegative) real num-ber ||v|| is called the magnitude or length of v. If v and w are two elementsRn, then ||v − w|| is called the distance between v and w. If v, w ∈ Rn, thenv ⊥ w if (v|w) = 0, in which case we say that v and w are orthogonal.Proposition 4 If v, w ∈ Rn, then1. ||v + w||2= ||v||2+ ||w||2+ 2(v|w).2. If v ⊥ w, then ||v + w||2= ||v||2+ ||w||2.3. |(v|w)| ≤ ||v|| ||w||.4. ||v + w|| ≤ ||v|| + ||w||.Of these, the only difficult one is (3), the Cauchy-Schwartz ine quality. It iseasy to prove (4) from (3) and (1).Definition 5 A sequence of vectors (v1, . . . vm) in Rnis orthogonal if vi⊥ vjwhenever i 6= j. The sequence is orthonormal if it is orthogonal and inaddition ||vi|| = 1 for all i.The following result is probably the most important theorem about innerproducts.Theorem 6 Let W be a linear subspace of Rnand let v be a member ofRn. There v can be written uniquelyv = πW(v) + π⊥W(v),where πW(v) ∈ W and π⊥W(v) is orthogonal to every vector in W . Further-more:1. πW(v) is the vector in W which is closest to v. That is,||πW(v) − v|| ≤ ||w − v|| for every w ∈ W ,with equality only if w = πW(v).2. If (w1, . . . wm) is an orthonormal basis for W , thenπW(v) = (v|w1)w1+ · · · (v|wm)wm.23. More generally, if (w1, . . . wm) if an orthogonal basis for W , thenπW(v) =(v|w1)w1||w1||2+ · · ·(v|wm)wm||wm||2.Let me explain a proof of (1). Let w0:= πW(v) This a vector in W . Let w beany other vector in W . We claim that if w 6= w0, then ||v − w|| > ||v − w0||.Let w0:= w0− w. This is another vector in W , and||v − w||2= ||(v − w0) + (w0− w)||2= ||π⊥(v) + w0||2= ||π⊥(v)||2+ ||w0||2since π⊥(v) is orthogonal to w0(use formula ). But ||w0||2≥ 0, and is zeroonly if w =
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