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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.

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
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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