Question

let x1,x2, and x3 be linearly independent vectors in R^(n) and let y1=x2+x1; y2=x3+x2; y3=x3+x1. are y1,y2,and y3 linearly independent? prove your answere?

Answer 1

Answer with Step-by-step explanation:

We are given that

`x_1,x_2` and
`x_3` are linearly independent.

By definition of linear independent there exits three scalar
`a_1,a_2` and
`a_3` such that

`a_1x_1+a_2x_2+a_3x_3=0`

Where
`a_1=a_2=a_3=0`

`y_1=x_2+x_1,y_2=x_3+x_2,y_3=x_3+x_1`

We have to prove that
`y_1,y_2` and
`y_3` are linearly independent.

Let
`b_1,b_2` and
`b_3` such that

`b_1y_1+b_2y_2+b_3y_3=0`

`b_1(x_2+x_1)+b_2(x_3+x_2)+b_3(x_3+x_1)=0`

`b_1x_2+b_1x_1+b_2x_3+b_2x_2+b_3x_3+b_3x_1=0`

`(b_1+b_3)x_1+(b_2+b_1)x_2+(b_2+b_3)x_3=0`

`b_1+b_3=0`

`b_1=-b_3`...(1)

`b_1+b_2=0`

`b_1=-b_2`..(2)

`b_2+b_3=0`

`b_2=-b_3`..(3)

Because
`x_1,x_2` and
`x_3` are linearly independent.

From equation (1) and (3)

`b_1=b_2`...(4)

Adding equation (2) and (4)

`2b_1==0`

`b_1=0`

From equation (1) and (2)

`b_3=0,b_2=0,b_3=0`

Hence,
`y_1,y_2` and
`y_3` area linearly independent.