Question

Let x1,x2 & x3 be linearly dependent vectors in R^n, let y1= x2-x1, y2=x3-x2,y3= x3-x1 are the y's linearly independetn?prove your answer

Answer 1

The y's should be linearly dependent. Assuming x1, x2 and x3 are linearly dependent, then we will have

c1x1 + c2x2 + c3x3 = 0, where not all the scalars are 0.

Now to check for the linear dependency of y.

c1y1 + c2y2 + c3y3 = 0; y1 = x2-x1, y2 = x3-x2, y3 = x3-x1

c1(x2-x1) + c2(x3-x2) + c3(x3-x1) = 0
c1x2 - c1x1 + c2x3 - c2x2 + c3x3 - c3x1 = 0
-(c1+c3)x1 + (c1-c2)x2 + (c2+c3)x3 = 0

That is equivalent to the equation c1y1 + c2y2 + c3y3 = 0

And since x1, x2, and x3 are linearly dependent, y1, y2 and y3 are linearly dependent. So no, y is NOT linearly independent.

If anyone has a different answer please correct me. I'm sure this is the right answer but I want to be sure