# Thread: Determine if W is a subspace of the vector space V.

1. ## Determine if W is a subspace of the vector space V.

Could someone help me out with questions (a) and (c), just the first couple of steps.

Any help is appreciated!

2. The axioms are scalar multiplication and vector addition, correct?

3. Originally Posted by dwsmith
The axioms are scalar multiplication and vector addition, correct?
Yes, let me write out the definitions:

V is the vector space

addition: If u & v are in V, then u + v is in V

multiplication: If k is any scalar & u is any object in V, then ku is in V

4. part c
$\displaystyle \alpha p(x)=\alpha(a+bx^2)=\alpha a + \alpha bx^2$ where $\displaystyle \alpha \in \mathbb{R}$

Therefore, $\displaystyle \alpha a + \alpha bx^2 \in V$

5. The key to the addition is that a and b aren't negative.

6. Originally Posted by dwsmith
The key to the addition is that a and b aren't negative.

k(a+bx^2) = ka + kbx^2

The k could be a negative number, making a and b negative, so the multiplication axiom wouldn't hold?

7. If k is negative, a and b are still positive.

ka=w would be negative.

8. The reason the problem says that is so this doesn't happen:

$\displaystyle a+bx^2+a+(-bx^2)=2a \not\in V$

9. Ok, that makes alot more sense, thank you!