-3 Bonus(5 pts), Let V be a vector space. Prove that for any pc V, oDm (no parti

Question

-3 Bonus(5 pts), Let V be a vector space. Prove that for any pc V, oDm (no partial credit). Vector Space Axioms Definition Let V' be a set on which the operations of addition and scalar multiplication ane defined. By this we mcan that, with each pair of elements x and y in V.we can associate a unique element x+ y that is also in V. and with cach element x in V and each scalar . we can associate a unique elements in 1.. The sci V, together with the operations of addition and scalar multiplication. is said to forn a vector space if the follow ing axioms are satisfied AL x + y y + x for any x and y in V A2. (x + y-+ z = x + (y + z) for any x. y, and z in 1, A3. There exists an element 0 in V such thatx+0x for each x E V A4. For each x e l'. there exists an element-x in V such that x + (-x) A5. ( x + y) ux t-uy for each scalar a and any x and y in V A6, ( + ) x = ux ÷ .f(r any scalars and and any x e l' A7. ()x = ( x, for any scalars and and any x e l. 0.

Explanation / Answer

let 0.v = z.

We want to show z = 0.

z = 0.v = (0+0).v = 0.v + 0.v (by A5) = z + z

Now, by A4, there exists an element -z such that z + (-z) = 0

Then (z + z) + (-z) = 0

by A2, z + ( z + (-z) ) = 0

So, z = 0

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