. If A ER is a common eigenvalue of two n x n matrices A and B, then A is also an eigenvalue of A+ B. If a couare matrix M has a able distribution then it must be regular
. If A ER is a common eigenvalue of two n x n matrices A and B, then A is also an eigenvalue of A+ B. If a couare matrix M has a able distribution then it must be regular
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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please send solution for Q9
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![For each question:
- Clearly state: True or False
- If True, provide a concise explanation
- If False, provide an explicit counterexample
1. Let A = b be a consistent, inhomogeneous system of linear equations. A solution to the system
is given by i = A-
2. Let S, T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly
dependent.
3. Let S C P(R) be the set of all even degree polynomials. Then S forms a subspace of P(R).
4. Consider two maps S, T : R" R". The composition SoT is a linear map if and only if both
S and T are linear maps
5. Any invertible 3 x 3 real matrix M is a matrix of change of basis [C]g for some bases B and C.
6. Let W be a finite dimensional subspace of an inner product space V. For each w e W, there
exists a unique v EV such that w is an orthogonal projection of v on W.
7. Let A be a 4 x 4 matrix such that the homogeneous equation Af = õ has general solution
i = Xữ for some non-zero vector TER. Then, rank(A) = 3.
%3D
8. Let V be a inner product space and let a e V be a non-zero vector. The subset U defined by
U = {ie V(F, ā) = 0}
is a subspace of V
9. If A ER is a common eigenvalue of two n x n matrices A and B, then A is also an eigenvalue
of A+ B.
10. If a square matrix M has a stable distribution, then it must be regular](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F861e3ce5-09b0-437a-ad8c-3c26f27a57e3%2Fe5c1c0fd-567e-4808-bb9b-d90737d73f10%2F3uqn9of_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For each question:
- Clearly state: True or False
- If True, provide a concise explanation
- If False, provide an explicit counterexample
1. Let A = b be a consistent, inhomogeneous system of linear equations. A solution to the system
is given by i = A-
2. Let S, T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly
dependent.
3. Let S C P(R) be the set of all even degree polynomials. Then S forms a subspace of P(R).
4. Consider two maps S, T : R" R". The composition SoT is a linear map if and only if both
S and T are linear maps
5. Any invertible 3 x 3 real matrix M is a matrix of change of basis [C]g for some bases B and C.
6. Let W be a finite dimensional subspace of an inner product space V. For each w e W, there
exists a unique v EV such that w is an orthogonal projection of v on W.
7. Let A be a 4 x 4 matrix such that the homogeneous equation Af = õ has general solution
i = Xữ for some non-zero vector TER. Then, rank(A) = 3.
%3D
8. Let V be a inner product space and let a e V be a non-zero vector. The subset U defined by
U = {ie V(F, ā) = 0}
is a subspace of V
9. If A ER is a common eigenvalue of two n x n matrices A and B, then A is also an eigenvalue
of A+ B.
10. If a square matrix M has a stable distribution, then it must be regular
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