1. Let V be a real vector space of finite dimension n ≥ 4, S = {v1, v2, v3} a subset of V and u ∈ If the set S ∪ {u} = {v1, v2, v3, u} is a generating set of the space V, it is correct to state that: a) the set S is linearly dependent; b) u ∈ [v1, v2, v3]; c) dim ([v1, v2] ∩ [v3, u]) ≥ 1; d) the set S ∪ {u} may or may not be linearly independent; e) u ≠ 0V; f) none of the above is correct. 2. Let A be a square matrix of order n whose characteristic polynomial is: pA(x) = x (x2 − 9)(x − 2)3 = x (x2 − 9)(x − 2)3. Rate as true (T) or false (F): a) The order of matrix A is n = 7 b) There is an eigenvector v ∈ IRn of A such that Av = 0. c) IRn has an eigenspace of dimension 1.
1. Let V be a real vector space of finite dimension n ≥ 4, S = {v1, v2, v3} a subset of V and u ∈ If the set S ∪ {u} = {v1, v2, v3, u} is a generating set of the space V, it is correct to state that: a) the set S is linearly dependent; b) u ∈ [v1, v2, v3]; c) dim ([v1, v2] ∩ [v3, u]) ≥ 1; d) the set S ∪ {u} may or may not be linearly independent; e) u ≠ 0V; f) none of the above is correct. 2. Let A be a square matrix of order n whose characteristic polynomial is: pA(x) = x (x2 − 9)(x − 2)3 = x (x2 − 9)(x − 2)3. Rate as true (T) or false (F): a) The order of matrix A is n = 7 b) There is an eigenvector v ∈ IRn of A such that Av = 0. c) IRn has an eigenspace of dimension 1.
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. Let V be a real vector space of finite dimension n ≥ 4, S = {v1, v2, v3} a subset of V and u ∈ If the set S ∪ {u} = {v1, v2, v3, u} is a generating set of the space V, it is correct to state that:
a) the set S is linearly dependent;
b) u ∈ [v1, v2, v3];
c) dim ([v1, v2] ∩ [v3, u]) ≥ 1;
d) the set S ∪ {u} may or may not be linearly independent;
e) u ≠ 0V;
f) none of the above is correct.
2. Let A be a square matrix of order n whose characteristic polynomial is:
pA(x) = x (x2 − 9)(x − 2)3 = x (x2 − 9)(x − 2)3.
Rate as true (T) or false (F):
a) The order of matrix A is n = 7
b) There is an eigenvector v ∈ IRn of A such that Av = 0.
c) IRn has an eigenspace of dimension 1.
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