Linear algebra Let S uv be linearly independent set Prove t

Let S = {u.v} be linearly independent set. Prove that the set {u + v, u - v} is linearly independent.

Solution

Let S = { u , v } is linearly independent set.

Now, let

a ( u + v ) + b ( u - v ) = 0

au +av + bu - bv = 0

combine terms of u and y, we getu

( a + b ) u + ( a - b ) v = 0

since u and v are linearly independent, so

a + b = 0 (1)

a - b = 0 (2)

On solving (1) and (2), we get

a = 0 and b = 0

Hence, { u + v, u - v } is linearly independent.