Question

True or False: Linear Equations in Linear Algebra

Answer 1

6. Consider each of the following statements and determine whether it is
true or false. Justify each answer.
(1) One needs at least 4 vectors in R4
to span R4
.
(2) Any set of 3 vectors in R4
is linearly independent.
(3) Let ~v1; ~v2; ~v3 be three linearly dependent vectors, then there must be
two of these three vectors such that one is a multiple of the other.
(4) Any set of 5 vectors in R4
is linearly dependent.
(5) The equation Ax = 0 always has a solution.
(6) If the equation Ax = 0 has only the trivial solution, then the columns
of A must be linearly dependent.
(7) If A is a 3 4 matrix, then columns of A must be linearly independent.
(8) If A is a 4 4 matrix and columns of A are linearly independent, then
A~x = ~
b is consistent for every
~
b in R4
.
(9) If ~v1; ~v2; ~v3; ~v4 are in R4
and ~v4 = ~v1+ ~v2+ ~v3, then the set f ~v1; ~v2; ~v3; ~v4g
is linearly independent.
(10) If A is a 34 matrix, then columns of A must be linearly independen

Answer 2

We aren't just going to do your homework for you. We will take your questions (usually one at a time), but only when it is clear you are trying to work it out and are stuck.

Answer 3

i should probably rephrase it then. its not homework, and i have the answer its review for a test. but a few of the problems i have no idea how my teacher justified it

Answer 4

1. True (not sure why)
2 False and Im guessing its because one of the vectors could be the zero vector
3. False, NO IDEA WHY
4. True, and its because there are more vectors than components
5. True, trivial solution
6. False, (not sure why)
7. False (not sure why)
8. True every column and row has a pivot
9. False, (not sure why)
10. False (not sure why)

So I guess im just asking for an explanation on a few of those. It was just easier for me to copy and paste the question