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Let n be a positive integer. Recall the standard inner product Y1 Y2 given as follows: If x = X1 x2 Xn , y = Yn 2 n then xy = xyi. A set of i=1 on R", 1, vectors u₁, U2,..., um in R" is orthonormal if u₁ · U₂ = [0, i = j i ‡ j . (a) For fixed y E R", prove that xx y gives a linear transformation from Rn to R. (b) Prove that an orthonormal set in R" is linearly independent. (c) If m is another positive integer and M is an m x n matrix with orthonormal columns, prove that (Mx).(My) = x.y for all x, y € Rn.

Let n be a positive integer. Recall the standard inner product Y1 Y2 given as follows: If x = X1 x2 Xn , y = Yn 2 n then xy = xyi. A set of i=1 on R", 1, vectors u₁, U2,..., um in R" is orthonormal if u₁ · U₂ = [0, i = j i ‡ j . (a) For fixed y E R", prove that xx y gives a linear transformation from Rn to R. (b) Prove that an orthonormal set in R" is linearly independent. (c) If m is another positive integer and M is an m x n matrix with orthonormal columns, prove that (Mx).(My) = x.y for all x, y € Rn.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let n be a positive integer. Recall the standard inner product on R", Y1 Y2 given as follows: If x = x1 X2 Xn , y = Yn 9 n then x y = xiyi. A set of • vectors U₁, U2, ..., um in R” is orthonormal if u₁ · uji = i=1 1, 0. i=j i ‡ j (a) For fixed y Є R", prove that x + xy gives a linear transformation from Rn to R. (b) Prove that an orthonormal set in R" is linearly independent. (c) If m is another positive integer and M is an m × n matrix with orthonormal columns, prove that (Mx).(My) = x-y for all x, y € R".
Transcribed Image Text:Let n be a positive integer. Recall the standard inner product on R", Y1 Y2 given as follows: If x = x1 X2 Xn , y = Yn 9 n then x y = xiyi. A set of • vectors U₁, U2, ..., um in R” is orthonormal if u₁ · uji = i=1 1, 0. i=j i ‡ j (a) For fixed y Є R", prove that x + xy gives a linear transformation from Rn to R. (b) Prove that an orthonormal set in R" is linearly independent. (c) If m is another positive integer and M is an m × n matrix with orthonormal columns, prove that (Mx).(My) = x-y for all x, y € R".
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