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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!

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1 Answer 1

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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.

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  • $\begingroup$ Thank you so much for your answer :D $\endgroup$
    – math254321
    Commented 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
    Commented Feb 1, 2018 at 0:23

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