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?

  1. Name the additive identity for each vectore space.

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

  2. 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!


1 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.

  • $\begingroup$ Thank you so much for your answer :D $\endgroup$
    – math254321
    Oct 1, 2014 at 15:56
  • $\begingroup$ One property of a vector space is that it must contain the "0" vector. Is that the case here? Another- if u is a third degree polynomial and v= -u, is u+ v a third degree polynomial? $\endgroup$
    – user247327
    Feb 1, 2018 at 0:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.