D EFINITION For u and v in R n the distance between u and v written as dist u v

D efinition for u and v in r n the distance between u

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D EFINITION 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 || D EFINITION Two vectors u and v in R n are orthogonal to each other if u · v = 0. T HEOREM 2: T HE P YTHAGOREAN T HEOREM Two vectors u and v are orthogonal if and only if || u + v || 2 = || u || 2 +|| v || 2 T HEOREM 3 Let A be an mxn 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 ) = Nul ( A ) and (Col A ) = Nul ( A T ) 26
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6.2 O RTHOGONAL S ETS D EFINITION A set of vectors u 1 ,..., u p in R n is said to be an orthogonal set if each pair of distinct vectors from the set is orthogonal, that is, if u i · u j = 0 whenever i 6= j T HEOREM 4 If S = u 1 ,..., u p 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 D EFINITION An orthogonal basis for a subspace W of R n is a basis for W that is also an orthogonal set. T HEOREM 5 Let u 1 ,..., u p be an orthogonal basis for a subspace W of R n . For each y in W , the weights in the linear combination y = c 1 u 1 + ... + c p u p are given by c j = y · u j u j · u j D EFINITION A set u 1 ,..., u p is an orthonormal set if it is an orthogonal set of unit vectors. If W is the subspace spanned by such a set, then u 1 ,..., u p is an orthonormal basis for W , since the set is automatically independent. T HEOREM 6 An m * n matrix U has orthonormal columns ⇐⇒ U T U = I . T HEOREM 7 Let U be an m * n matrix with orthonormal columns, and let x and y be in R n . Then, 1. || Ux || = || x || 2. ( Ux ) · ( U y ) = x · y 3. ( Ux ) · ( U y ) = 0 if and only if x · y = 0 D EFINITION An orthogonal matrix U is a useful SQUARE matrix that has the property U T = U - 1 . By definition, the columns AND rows are orthonormal. 27
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6.3 O RTHOGONAL P ROJECTIONS T HEOREM 8: T HE O RTHOGONAL D ECOMPOSITION T HEOREM Let W be a subspace of R n . Then each y in R n can be written uniquely in the form y = ˆ y + z where ˆ y is in W and z is in W . In fact, if u 1 ,..., u p is any orthogonal basis in W , then ˆ y = y · u 1 u 1 · u 1 u 1 + ... + y · u p u p · u p and z = y - ˆ y P ROPERTY OF O RTHOGONAL P ROJECTIONS If y is in W = Span( u 1 ,..., u p ), then proj W y = y . T HEOREM 9: T HE B EST A PPROXIMATION T HEOREM Let W be a subspace of R n . let y be any vector in R n , and let ˆ y be the orthogonal projection of y onto W . Then ˆ y is the closest point in W to y , in the sense that || y - ˆ y || < || y - v || for all v in W distinct from ˆ y . T HEOREM 10 If u 1 ,... u p is an orthonormal basis for a subspace W of R n , then proj W y = ( y · u 1 ) u 1 + ( y · u 2 ) u 2 + ... + ( y · u p ) u p If U = £ u 1 u 2 ... u p / , then proj W y = UU T y for all y in R n 28
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6.4 T HE G RAM -S CHMIDT P ROCESS T HEOREM 11: T HE G RAM -S CHMIDT P ROCESS Given a basis x 1 ,..., x p or a nonzero subspace W of R n , define v 1 = x 1 v 2 = x 2 - x 2 · v 1 v 1 · v 1 v 1 v 3 = x 3 - x 3 · v 1 v 1 · v 1 v 1 - x 3 - x 3 · v 2 v 2 · v 2 v 2 ...... v p = x p - x p · v 1 v 1 · v 1 v 1 - x p - x p · v 2 v 2 · v 2 v 2 - ... - x p · v p - 1 v p - 1 · v p - 1 v p - 1 Then, { v 1 ,..., v p } is an orthogonal basis for W . In addition, Span{ v 1 ,..., v k } = Span{ x 1 ,..., x k } for 1 k p T HEOREM 12: T HE QR F ACTORIZATION If A is an m x n matrix with linearly independent columns, then A can be factored as A = QR where Q is an m x n
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