- #1

mosawi

- 9

- 0

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. If

**u**and

**v**are objects in

**V**, then

**u+v**is in

**V**

2.

**u+v = v + u**

3.

**u+(v+w) = (u+v)+w**

4. There is an object 0 in

**V**, called a

**zero vector**for

**V**, such that

**0+u = u+0 = u**for all

**u**in

**V**

5. For each

**u**in

**V**, there is an object

**-u**in

**V**, called a negative of

**u**, such that

**u+(-u) = (-u)+u = 0**

6. If

**k**is any scalar and

**u**is any object in

**V**, then

**ku**is in

**V**

7.

**k(u+v) = ku+kv**

8.

**(k+m)u = ku+mu**

9.

**k(mu) = (km)(u)**

10.

**1u = u**

Definition of subspace:

A subset

*W*of a vector space

*V*is called a

**subspace**of

*V*if

*W*is itself a vector space under the addition and scalar multiplication defined on

*V*.

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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