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?
Name the additive identity for each vectore space.
a) 6-Space (R^6) b) M4,3
Determine whether the given set with standard operations is a subspace of a known vector space. Identify the vector space.
<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!