show that if v1 v2 v3 are linearly independent then the vect

show that if v₁, v₂, v₃ are linearly independent, then the vectors w₁ = v₁ + v₂, w₂ = v₁ + v₃, and w₃ = v₂ +v₃ are also linearly independent.

Solution

we need to prove that if v1, v2, v3 are linearly independent, then the vectors w1 = v1 + v2, w2 = v1 + v3, and w3= v2 +v3 are also linearly independent.

we know that vectors v1 , v2 ,v3 are linearly independent if and only if some linear combination of v1 , v2 , v3 adds up to 0, it turns out that c1, c2, c3 are all zero

means v1 , v2 , v3 are linearly independent if

c1v1 + c2v2 + c3v3 = 0 and c1 = c2 = c3 = 0

simillarly vectors w1 , w2 , w3 are linearly independent if d1w1 + d2w2 + d3w3 = 0 while d1 = d2 = d3 = 0

we have,

w1 = v1 + v2

w2 = v1 + v3

w3 = v2 + v3

we can write,

d1(v1+v2) + d2(v1+v3) + d3(v2+v3) = 0

if we proved that d1 =d2 = d3 = 0 then we can say that w1 , w2 , w3 are linearly independent

rearranging we have,

(d1+d2)v1 + (d1+d3)v2 + (d2+d3)v3 = 0

Because v1, v2, v3 are linearly independent the coefficient of v1 , v2 , v3 must be 0

so we can say that,

d1 + d2 = 0 -------------1)

d1 + d3 = 0 ---------------2)

d2 + d3 = 0 ----------------3)

subtract 2) from 1) we have,

d1 + d2 - d1 - d3 = 0

d2 - d3 = 0 ------------4)

subtract 3) from 4) we have,

d2 - d3 - d2 - d3 = 0

-2d3 = 0 so d3 = 0

put d3 = 0 in 4) we have,

d2 - 0 = 0 so d2 = 0

put d2 = 0 in 1) we have

d1 + 0 0 so d1 =0

hence we have d1 = d2 = d3 = 0

we can say that w1, w2 , w3 are linearly independent