Question

Determine if the given set constitutes Real Vector Space (A)The Set of all elements of R^3 with first component of 1 (B) The set of all non singular 2X2 matrices with real elements

Answer 1

In both cases at least one axiom is violated.

Answer 2

Could you go into detail about how you arrived at this answer?

Answer 3

Well, a subspace of a vector space is a vector space if it satisfies the axiom of a vector space.

So list out those axioms, and check each one to see if they are satisfied for these subspaces. I think you'll find in short order they are not.

Answer 4

Notice also to prove they are not vector spaces, it is sufficient to show one axiom is not satisfied.

Answer 5

For example, here's one axiom. For all \( v, w \in V \) then \( v + w \in V \). Is that axiom satisfied here?

Answer 6

Is this abstract algebra or introductory proof-based mathematics?

Answer 7

This is for a differential Equations class, I've been away from math for a while so the more abstract stuff is confusing.

Answer 8

I think that axiom would not be satisfied for part A of my question

Answer 9

No, it's not, because for any two vectors thus described, there sum is
(1,a,b) + (1,c,d) = (2,a+c,b+d)
which is not such a vector

Answer 10

*their

Answer 11

Other axioms are violated as well. For example, is the zero vector a member of that set? And what about scalar multiples? Or additive identities?

Answer 12

I think the zero vector should be satisfied because u+0=u right?

Answer 13

No, because the zero vector isn't a member of the set.

Answer 14

is there a good reference on this I seem to be getting further away from this?

Answer 15

I'd ask your professor. But any good text book on linear algebra should do the trick.

Answer 16

http://www.khanacademy.org/#linear-algebra
http://www.khanacademy.org/#differential-equations

But, yeah, ask your professor.

Answer 17

ok thank you both