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1) a) State and verify the result of the Cauchy-Schwarz inequality from Chapter 3 for the vectors: u = (1, -5,3), v = (2,1, -2). b) Determine whether the angle between these vectors is acute, obtuse, or right.

1) a) State and verify the result of the Cauchy-Schwarz inequality from Chapter 3 for the vectors: u = (1, -5,3), v = (2,1, -2). b) Determine whether the angle between these vectors is acute, obtuse, or right.

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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1) a) State and verify the result of the Cauchy-Schwarz inequality from Chapter 3 for the vectors: u = (1, -5,3), v = (2,1,-2). b) Determine whether the angle between these vectors is acute, obtuse, or right. 2) Find a vector that is perpendicular to both vectors in Problem 1 and whose length is equal to 2. Is there any other vector satisfying these conditions? 3) For the vectors from Problem 1, find the vector component of u along v. 4) Determine whether the set of all 3 x 3 matrices A over R with det (A) = 0 is a vector subspace of the vector space of all 3 x 3 matrices. 5) Determine whether the set {(2, 0, 1), (1, 1, 0), (0, 0, 1)} spans R3. Is this set linearly independent? 6) W = {p € Pz. | p(1) = 0) is a subspace of P2 (you do not need to show this, you may assume it). Find a basis for W and use your answer to determine the dimension of W. 7) Give an example of a set S with a binary operation on S and a scalar multiplication on S such that S is closed under the binary operation and the scalar multiplication but is not a vector space with respect to them.
Transcribed Image Text:1) a) State and verify the result of the Cauchy-Schwarz inequality from Chapter 3 for the vectors: u = (1, -5,3), v = (2,1,-2). b) Determine whether the angle between these vectors is acute, obtuse, or right. 2) Find a vector that is perpendicular to both vectors in Problem 1 and whose length is equal to 2. Is there any other vector satisfying these conditions? 3) For the vectors from Problem 1, find the vector component of u along v. 4) Determine whether the set of all 3 x 3 matrices A over R with det (A) = 0 is a vector subspace of the vector space of all 3 x 3 matrices. 5) Determine whether the set {(2, 0, 1), (1, 1, 0), (0, 0, 1)} spans R3. Is this set linearly independent? 6) W = {p € Pz. | p(1) = 0) is a subspace of P2 (you do not need to show this, you may assume it). Find a basis for W and use your answer to determine the dimension of W. 7) Give an example of a set S with a binary operation on S and a scalar multiplication on S such that S is closed under the binary operation and the scalar multiplication but is not a vector space with respect to them.
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