Hello all- I'm returning to the Maths after a hiatus and I'm already regretting it Very basic question

determine whether each set with the given operations is a vector space. If not ID the axiom that fails

for a_{0 + }a_{1}x with the operations

(a_{0}+ a_{1}x) + (b_{0}+ b_{1}x) = (a_{0}+ b_{0}) + (a_{1}+ b_{1})x

and

k(a_{0}+ a_{1}x) = (ka_{0}) + (ka_{1})x

-----------------------------------------------------

My original thought is that x is a scaler.......but maybe not= a0, a1x

uv= b0,b1x

+

uv=v+u

a0 + b0 = b0 + a0

a1x + b1x = b1x + a1x = (a1+b1) x

so closed under addition

the sencond part looks fine to me so closed under multiplication because the scaler can be applied to each component individually.

I'm having trouble with the Zero vector (which can be 0 or anything else????) For the zero vector if a0 or a1 are zero than multiplying by any c will give 0 so it passes.

-----------------------------------------------------------

The follow up Question True or False

The set of polynomials with degree exactly 1 is a vector space under the operations defined in exercise 12?

Back of the book Answer says false????????????? I know I'm blowing it somewhere (probably everywhere) please help nudge me in the right direction.

Anthony