Question

Determine whether the following subsets of R^3 are subspaces:
Vectors of the form (a,b,2)
Vectors of the form (a, b,c ) where c=a+b.
Not sure how to test these out, how do I know these are non-empty, and then check the two rules?

Answer 1

do you know what you need to check?

Answer 2

first off you need that the zero vector is in your subspace. in \(\mathbb{R^3}\) the zero vector is \(<0,0,0>\) so you can forget about the first one

Answer 3

the second one contains the zero vector, so you have to check the other conditions, namely that it is closed under linear combinations and scalers. in other words you have to check first of all that if
\(\overrightarrow{u}=<a_1,b_1,a_1+b_1>\) and \(\overrightarrow{v}=<a_2,b_2,a_2+b_2>\) are two such vectors, that \(c\overrightarrow{u}+d\overrightarrow{v}\)is of the same form

Answer 4

grind it out and you will see that it is

Answer 5

hope it is clear what you have to do, just write it out and say "look, this is also of the form \(<a,b,a+b>\)"

Answer 6

I'm working on the second one now

Answer 7

ok let me know if it is not clear

Answer 8

Okay, I got the second one. What about polynomials? For example, polynomials of the form a1+ a2x+ a3x^2, where a1=0 or a1=2.

Answer 9

this one is not closed under addition i think
if you add to polynomials with constant 2, the constant is now 4

Answer 10

awesome thanks!