Let V be a vector space and S be a subspace of V Define a re

Let V be a vector space and S be a subspace of V. Define a relation on V as x y if and only if x - y S. Prove that is an equivalence relation on V. For V - R^2, S = {x = (x_1, x_2) R^2 : x_1 = 0}, determine all of the equivalence classes of.

Solution

i)

1. Check for reflexivity

x~x as:x-x=0 and 0 belongs to S as S is a subspace.

2. Check for symmetry

x~y means x-y is in S. But S is a vector space so:-(x-y) is in S ie y-x is in S

Hence, y~x

HEnce, ~ is symmetric

3. Check for transitive

x~y,y~z hence, x-y is in S and y-z is in S

SO, x-y+y-z=x-z is in S

Hence, x~z

Hence, ~ is transitive.

ii)

LEt, x=(r1,r2)

y=(t1,t2)

x-y is in S hence, r2-t2=0

or r2=t2

So an equivalence class is: E_a={(x,a): x\\in R , a is some real number}

We have such an equivalence class for each real number, a