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I hope you guys can help me. I am soo confused with this question: I would really liked a complete answer to this, I have an upcoming exam and I know these two will be on the exam.

1. Let V be the set of all diagonal 2x2 matrices i.e. V = {[a 0; 0 b] | a, b are real numbers} with addition defined as A ⊕ B = AB, normal scalar multiplication. Prove all 10 axioms. Is V a vector space? If it is not a vector space, which axiom(s) fail?

2. Let W = {(x,y) ∈ ℝ^2 | x^2 + y^2 <= 1}. Show that W is not a subspace of ℝ^2.

The 10 axioms are:

1. Ifuandvare objects inV, thenu+vis inV

2.u+v = v + u

3.u+(v+w) = (u+v)+w

4. There is an object 0 inV, called azero vectorforV, such that0+u = u+0 = ufor alluinV

5. For eachuinV, there is an object-uinV, called a negative ofu, such thatu+(-u) = (-u)+u = 0

6. Ifkis any scalar anduis any object inV, thenkuis inV

7.k(u+v) = ku+kv

8.(k+m)u = ku+mu

9.k(mu) = (km)(u)

10.1u = u

Definition of subspace:

A subsetWof a vector spaceVis called asubspaceofVifWis itself a vector space under the addition and scalar multiplication defined onV.

I may be wrong by this but to prove something is a subspace, we must only show it passes axiom 1 and axiom 6, because all the other axioms are inherited.

I totally understand all the other review questions except these two, any help or advice will be very much appreciated, thanks.

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# Is V a vector space? and Show that W is not a subspace

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