5. For each of the statements given below decide if it is true or false. If it is true explain why. If it is false give a counterexample. a) If A and B are square matrices such that det(AB) = 1 then both matrices A and B must be invertible. b) If A is a 3 x 3 matrix and Col(A) is its column space, then dim Col(A) = 3. 2 c) If B = {V1, V2, V3} is a basis of R³ and u is a vector in R³ such that (4-9 [u] B = then u must be in Span(V2, V3).
5. For each of the statements given below decide if it is true or false. If it is true explain why. If it is false give a counterexample. a) If A and B are square matrices such that det(AB) = 1 then both matrices A and B must be invertible. b) If A is a 3 x 3 matrix and Col(A) is its column space, then dim Col(A) = 3. 2 c) If B = {V1, V2, V3} is a basis of R³ and u is a vector in R³ such that (4-9 [u] B = then u must be in Span(V2, V3).
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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Question
![5. For each of the statements given below decide if it is true or false. If it is true explain
why. If it is false give a counterexample.
a) If A and B are square matrices such that det(AB) = 1 then both matrices A and B must
be invertible.
b) If A is a 3 x 3 matrix and Col(A) is its column space, then dim Col(A) = 3.
2
c) If B = {V1, V2, V3} is a basis of R³ and u is a vector in R³ such that
[u] B
=
then u must be in Span(V₂, V3).
d) If V is a subspace of R³ and u, v € R³ are vectors such that u + v € V and u-VE V
then u € V.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7d69634b-8a8a-4609-8704-3bdadaefe256%2F1c16e54b-64cd-4d26-9bb6-b60ed1b94004%2F9vencs_processed.png&w=3840&q=75)
Transcribed Image Text:5. For each of the statements given below decide if it is true or false. If it is true explain
why. If it is false give a counterexample.
a) If A and B are square matrices such that det(AB) = 1 then both matrices A and B must
be invertible.
b) If A is a 3 x 3 matrix and Col(A) is its column space, then dim Col(A) = 3.
2
c) If B = {V1, V2, V3} is a basis of R³ and u is a vector in R³ such that
[u] B
=
then u must be in Span(V₂, V3).
d) If V is a subspace of R³ and u, v € R³ are vectors such that u + v € V and u-VE V
then u € V.
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