The Generalized Gram-Schmidt Process can be used to find an orthogonal basis for any subspace spanned by a finite linearly independent subset. If B = ( v 1, v 2,…, v k) is an orthogonal ordered basis for a subspace W of an inner product space V, and if v is any vector in W, then B =. ■Īn orthogonal set of nonzero vectors in an inner product space is a linearly independent set.
Orthogonal and orthonormal sets of vectors, and orthogonal complements of subspaces, are defined for inner product spaces analogously as for real vector spaces. The angle θ between two vectors in a real inner product space is defined as the angle between 0 and π such that cos θ = 〈 x, y 〉 / ( ‖ x ‖ ‖ y ‖ ). The length of a vector x in an inner product space is ‖ x ‖ = 〈 x, x 〉, and the distance between vectors x and y in an inner product space is | | x − y | |.
■įor vectors x, y and scalar k in a complex inner product space, 〈 x, y 〉 = 〈 y, x 〉 ¯, 〈 x, k y 〉 = k ¯ 〈 x, y 〉, and ‖ k x ‖ = | k | ‖ x ‖. ■įor vectors x, y and scalar k in a real or complex inner product space, 〈 k x, y〉 = k 〈 x, y〉. ■įor vectors x, y and scalar k in a real inner product space, 〈 x, y〉 = 〈 y, x〉, and 〈 x, k y〉 = k 〈 x, y〉. ■Īn inner product space is a vector space that possesses three operations: vector addition, scalar multiplication, and inner product. Real and complex inner products are generalizations of the real and complex dot products, respectively.
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