Linear Algebra: Vector Space, Standard Operation

Question

I have some questions about linear algebra.

1.Determine whether the set of all third degree polynomials with standard operations is a vector space. If it is not, identify each of the vector space axioms that fail.

These are the axioms.

u+v is in V. Closure under addition.

u+v=v+u. Commutative property.

u+(v+w)=(u+v)+w. Associative property.

V has a zero vector 0 such that for every u∈V, u+0=u. Additive identity.

For every u∈V, there is a vector in V denoted by −u such that u+(−u)=0. Additive inverse.

cu is in V. Closure under scalar multiplication.

c(u+v)=cu+cv. Distributive property.

(c+d)u=cu+du. Distributive property.

c(du)=(cd)u. Associative property.

1(u)=u. Scalar identity.

I am thinking the answers are Closure under addition, Additive identity, and Closure under scalar multiplication.

Is that correct? If it is, how can I prove that those fail?

2. Name the additive identity for each vectore space.

   a) 6-Space (R^6) b) M4,3

3. Determine whether the given set with standard operations is a subspace of a known vector space. Identify the vector space.

a) W=<3s+2t, 2s-t, t>: s, t are elements of R

b) The set of all 3 x 3 diagonal matrices

• R stands for all real numbers.

Please help me!

Answer 1

You're right that the way to solve the first one is to check the axioms individually. Make a couple of third degree polynomials $u(x) = ax^3 + bx^2 + cx + d$ and $v(x) = ex^3 + fx^2 + gx + h$ and play around with them.

What is $u(x) + v(x)$? Is it a third degree polynomial? If so, you have closure under addition.

What is the additive identity? Is that a third degree polynomial?

What do you get when you multiply $u(x)$ by a scalar $k$? Is that a third degree polynomial?

Et cetera.