Chapter 6 notes - 6.1 Inner Product Length and Orthoganlity Theorem 1 Let u,v and w be vectors in Rn and let c be a scalar Then a uv = vu b(u v w = uw

# Chapter 6 notes - 6.1 Inner Product Length and Orthoganlity...

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6.1 Inner Product, Length, and Orthoganlity Theorem 1 Let u,v, and w be vectors in R n , and let c be a scalar. Then a. u∙v = v∙u b. (u+v) ∙w = u∙w+v∙w c. ( c u) ∙v= c (u∙v)=u∙ (c v) d. u∙u≥0, and u∙u = 0 if and only if u = 0 Definition The length(or norm) of v is the nonnegative scalar ||v|| defined by ||v|| = sqrt(v ∙v) = sqrt(v1 2 +…vn 2 ) and ||v|| 2 =v ∙v Definition For u and v in R n , the distance between u and v, written as dist(u,v), is the length of the vector u-v. That is, dist(u,v) = ||u-v|| Definition Two vectors u and v in R n are orthogonal (to each other) if u∙v = 0 Theorem 2 The Pythagorean Theorem Two vectors u and v are orthogonal if and only if ||u+v|| 2 =||u|| 2 +||v|| 2 Theorem 3 Let A be an m x n matrix. The orthogonal complement of the row space of A is the null space of A, and the orthogonal complement of the column space of A is the null space of A T : (Row A) perp =Nul A and (Col A) perp = Nul A T
6.2 Orthogonal Sets Theorem 4 If S = {u1,…,up} is an orthogonal set of nonzero vectors in R n , then S is linearly independent and hence is a basis for the subspace spanned by S.

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• Spring '08
• Chorin