How do I prove this? Let V be a nonemp an addition operation and a scalar multip

Question

How do I prove this?

Let V be a nonemp an addition operation and a scalar multiplication operation with scalars V a vector space over F, provided the following ten pty set (whose elements are called vectors) on which are defined in F. We call conditions are satisfied: AL. Closure under addition: For each pair of vectors u and v in V, the sum u under addition: For each pair of vectors u and v in V, the sumu+v is also in V. We say that V is closed under addition. 2. Closure under scalar multiplication: For each vector v in V and each scalar k n F, the scalar multiple kv is also in V. We say that V is closed under scalar multiplication A3. Commutativity of addition: For all u, v V, we have u+v=v+u. A4. Associativity of addition: For all u, v, w e V, we have (u + v) + w = u + (v + w). A5. Existence of a zero vector in V: In V there is a vector, denoted 0, satisfying v+0=v, for all v E V. A6. Existence of additive inverses in V: For each vector v In V, there is a vector denoted -V, in V such that v+1-v) = 0. A7. Unit property: For all veV Iv=v. A8. Associativity of scalar multiplication: For all v e V and all scalars r, sEF. (rs)vr(sv). A9. Distributive property of scalar multiplication over vector addition: For all u V E V and all scalars r EF r(u + v) = ru + rv. A10. Distributive property of scalar multiplication over scalar addition: For all v e V and all scalars r,SEF (r+s)v = rv + sv.

Explanation / Answer

A1. Closure under addition :

For each real number x and y, sum x + y = xy = Real number, so, it is closed under addition

A2. Closure under scalar multiplication :

For each real number x and scalar c, their product cx = xc , which is also real, so it is closed under scalar multiplication.

A3. Commutativity of addition :

x + y = xy = yx = y + x

A4. Associativity of addition :

(x + y) + z = xy + z = xyz = x + yz = x + (y + z)

A5. Existence of additive identity :

x + 1 = 1x = x

and 1 is a real number, so Additive identity exists.

A6. Existence of Additive inverse :

For each real number x, 1/x also exists such that x + 1/x = x (1/x) = 1

A7. Unit property :

1x = x1 = x

A8. Associativity of scalar multiplication :

For a real number x, and scalars c and d,

(cd)x = xcd = (xd)c = cxd = c(dx)

A9. Distributive property of scalar multiplication over vector addition :

For vectors x and y, and scalar c,

c(x + y) = c(xy) = (xy)c = xcyc = (cx)(cy) = cx + cy

A10. Distributive property of scalar multiplication over scalar addition :

For vector x, and scalars c and d,

(c + d)x = (cd)x = xcd = xcxd = xc + xd = cx + dx.

Hence, it is a vector space.

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