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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?

User Rgahan
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1 Answer

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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.

User Hardik Kothari
by
7.7k points

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