10. (Strang 4.1.28) Explain why each of these statements is false: (a) ā = [1,1, 1]" is perpendicular to b = [1, 1, –2]", so the planes d"r = x + y + z = 0 and b"r = x + y – 2z = 0 are orthogonal subspaces of R³. (b) The subspace of R' spanned by (1, 1,0,0, 0) and (0,0,0, 1, 1) is the orthogonal com- plement of the subspace spanned by (1, – 1,0, 0, 0) and (2, –2, 3, 4, –4). (c) Two subspaces that meet only at the zero vector must be orthogonal.
10. (Strang 4.1.28) Explain why each of these statements is false: (a) ā = [1,1, 1]" is perpendicular to b = [1, 1, –2]", so the planes d"r = x + y + z = 0 and b"r = x + y – 2z = 0 are orthogonal subspaces of R³. (b) The subspace of R' spanned by (1, 1,0,0, 0) and (0,0,0, 1, 1) is the orthogonal com- plement of the subspace spanned by (1, – 1,0, 0, 0) and (2, –2, 3, 4, –4). (c) Two subspaces that meet only at the zero vector must be orthogonal.
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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![10. (Strang 4.1.28) Explain why each of these statements is false:
(a) ā = [1, 1, 1]" is perpendicular to b = [1, 1, –2]", so the planes a7 = x + y + z = 0
and b"r = x + y – 2z = 0 are orthogonal subspaces of R³.
(b) The subspace of R spanned by (1, 1, 0, 0, 0) and (0,0,0, 1, 1) is the orthogonal com-
plement of the subspace spanned by (1, – 1,0, 0,0) and (2, –2, 3, 4, –4).
(c) Two subspaces that meet only at the zero vector must be orthogonal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d4660a8-338a-4a16-b784-bd05cacd2eee%2Fff231886-1a33-4599-91ae-1033be163fa8%2Fluvsijh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:10. (Strang 4.1.28) Explain why each of these statements is false:
(a) ā = [1, 1, 1]" is perpendicular to b = [1, 1, –2]", so the planes a7 = x + y + z = 0
and b"r = x + y – 2z = 0 are orthogonal subspaces of R³.
(b) The subspace of R spanned by (1, 1, 0, 0, 0) and (0,0,0, 1, 1) is the orthogonal com-
plement of the subspace spanned by (1, – 1,0, 0,0) and (2, –2, 3, 4, –4).
(c) Two subspaces that meet only at the zero vector must be orthogonal.
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