(a) Let V = Rª and let 3 be the standard dot product on V. (i) Calculate B((1,0, 1,0), (0, 1,0, 1)). (ii) If u = (x1,..., T4) E V, show that B(u, u) = u · u > 0. (iii) Give an example of a vector of length 3. (iv) If u = (1,0, 1,0) e V, find S = {v € V : B(u, v) = 0} and state the dimension of S. (b) If V = R", and U is a subspace of V of dimension m with 0 < m < n, prove that υ' - (vεV : β(u, υ) 0 for all ueU; is a subspace of V, and state (using a theorem from class if needed) its dimension. (c) If V = R² define 3 B((x1, x2), (yı, Y2)) := (x1, x2). and show that 3 is a positive definite scalar product on V. (d) Use the Gram-Schmidt process to find an orthonormal basis of R³ that contains a scalar
(a) Let V = Rª and let 3 be the standard dot product on V. (i) Calculate B((1,0, 1,0), (0, 1,0, 1)). (ii) If u = (x1,..., T4) E V, show that B(u, u) = u · u > 0. (iii) Give an example of a vector of length 3. (iv) If u = (1,0, 1,0) e V, find S = {v € V : B(u, v) = 0} and state the dimension of S. (b) If V = R", and U is a subspace of V of dimension m with 0 < m < n, prove that υ' - (vεV : β(u, υ) 0 for all ueU; is a subspace of V, and state (using a theorem from class if needed) its dimension. (c) If V = R² define 3 B((x1, x2), (yı, Y2)) := (x1, x2). and show that 3 is a positive definite scalar product on V. (d) Use the Gram-Schmidt process to find an orthonormal basis of R³ that contains a scalar
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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Transcribed Image Text:4. (a) Let V = R* and let B be the standard dot product on V.
(i) Calculate B((1,0, 1,0), (0, 1,0, 1)).
(ii) If u = (x1,..., x4) E V, show that B(u, u) = u · u > 0.
(iii) Give an example of a vector of length 3.
(iv) If u = (1,0, 1,0) e V, find
S = {v € V : B(u, v) = 0}
and state the dimension of S.
(b) If V = R", and U is a subspace of V of dimension m with 0 < m< n, prove that
U- = {v € V : B(u, v) = 0 for all u E U}
%3D
is a subspace of V, and state (using a theorem from class if needed) its dimension.
(c) If V = R² define
3 1
Y1
В(21, т2), (ул, у2)) :%3 (21, х).
1 4
Y2
and show that ß is a positive definite scalar product on V.
(d) Use the Gram-Schmidt process to find an orthonormal basis of R³ that contains a scalar
multiple of the vector v1 = (1,0, –1).
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