Show that in vector spaceV(F) ,xv = yv ; implies x = y. where , x, y belongs to F( field) & v belongs to V .?

Question

V IS VECTOR SPACE.

Answer 1

First, the statement as it stands is not true: if v is the zero vector, then x need not equal y. What is true is that  xv=yv implies that either v is the zero vector or x=y. Proof:

add  -yv to both sides:

xv-yv = 0.

By the distributive law (i.e., factor):

(x-y)v = 0.

Now one possibility is that x-y=0. In that case, add y to both sides to get x=y.

If (x-y) is not zero , then we use the simple Thm in Linear algebra that states: if kv=0 and and k is not =0, then v is the zero vector.** Letting k=x-y then yields that v must = 0. So we see that xv=yv need not imply that x=y if v is the zero vector.

[** proof of this goes as follows: if k is not =0, then by the field properties, it has an inverse k^(-1) such that k^(-1) * k = 1.

Then:

v = 1v  (vector space axiom)

= [k^(-1) * k]v = [k^(-1)]*(kv)   (associative law)

= k*0  (since kv=0)

= 0.

Now let k=x-y for our problem ]