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Orthogonality in Rn Math 214 { Spring, 2013 Apr 22 Orthogonality in Rn n A set of vectors fv1; v2;::: vkg in R is called an orthogonal set if all the vectors in the set are pairwise orthogonal, that is, vi · vj = 0 for all i 6= j, i; j = 1; 2; : : : ; k. Example: Check whether the following vectors form an orthogonal set: 2 1 3 2 2 3 2 1 3 4 2 5 ; 4 0 5 ; 4 −1 5. −1 2 −1 n A basis for a subspace W of R that forms an orthogonal set is called an orthogonal basis. Orthogonal bases are more convenient, as it is easier to find the coefficients in the expansion of vectors in terms of the basis vectors if it is an orthogonal basis. n Theorem: Let fv1; v2;::: vkg be an orthogonal basis for a subspace W of R , and let w be an vector in W . The the unique scalars in the expansion of w, w = c1v1 + c2v2 + ··· + ckvk are given by w · vi ci = for i = 1; 2; : : : ; k: vi · vi It's even more convenient to work with an orthogonal basis in which all the vectors are unit vectors. n A set of vectors in R is called an orthonormal set if it is an orthogonal set of unit vectors. n An orthonormal basis for a subspace W of R is a basis of W which is an orthonormal set. n Theorem: Let fq1; q2;::: qkg be an orthonormal basis for a subspace W of R , then any vector w of W has the unique expansion w = (w · q1)q1 + (w · q1)q2 + ··· + (w · qk)qk: Orthogonal matrices Theorem: The columns of an m × n matrix Q form an orthonormal set if and only if T Q Q = In. An n×n matrix Q whose columns form and orthonormal set is called and orthogonal matrix. From the previous Theorem it is clear that: Theorem: A square matrix Q is orthogonal if an only if Q−1 = QT . Example: The standard matrices of rotations are orthogonal (why?): cos θ − sin θ A = . sinθ − cos θ 1 Theorem: Let Q be an n × n matrix, then the following statements are equivalent. (a) Q is orthogonal. n (b) jQxj = jxj for every x in R . n (c) Qx · Qy = x · y for every x and y in R . From the definition of an orthogonal matrix, it is clear that. Theorem: The rows of an orthogonal matrix form an orthonormal set. Here are some of the properties of orthogonal matrices. Theorem: Let Q be an orthogonal matrix, then (a) Q−1 is orthogonal. (b) det Q = ±1. (c) If λ is an eigenvalue of Q, then jλj = 1. (d) If Q1 and Q2 are orthogonal n × n matrices, then so is Q1Q2. Orthogonal complements and projections n n Let W be a subspace of R . A vector v in R is orthogonal to W , if v is orthogonal to every vector in W . The set of all vectors orthogonal to W is called the orthogonal complement of W and is denoted by W ?, ? n W = fv in R : v · w = 0 for all w in W g: n Theorem: Let W be a subspace of R , then ? n (a) W is a subspace of R . (b) (W ?)? = W . (c) W \ W ? = f0g. ? (d) If W = span(w1;:::; wk), then v is in W if and only if v · wi = 0 for all i = 1; : : : ; k. For the subspaces associated with an orthogonal matrix we have the following result. Theorem: Let A be an orthogonal m × n matrix, then the orthogonal complement of row(A) is null(A) and the orthogonal complement of col(A) is null(AT ). Similar to projection onto the direction of a vector, we can define projections onto n subspaces. Let W be a subspace of R and let fu1; u2;::: ukg be an orthogonal basis for n W . For any vector v in R the orthogonal projection onto W is v · u1 v · u2 v · uk projW v = u1 + u2 + ··· + uk: u1 · u1 u2 · u2 uk · uk 2 n Theorem (Orthogonal decomposition): Let W be a subspace of R and let v be any vector n ? ? in R . Then there are unique vectors w in W and w in W such that v = w + w?: As a corollary from the previous theorem we have. n ? Theorem: If W is a subspace of R , then dim W + dim W = n. Exercises: 2 2 3 2 4 3 1. Let W be a subspace of R3 spanned by the vectors 4 1 5, 4 0 5. Find a basis for −2 1 W ?. 3.
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