Are x1, x2, and x3 linearly dependent in R^n?

[DMOC]

y1 = x2 - x1

y2 = x3 - x2

y3 = x3 - x1

x1, x2, x3 are linearly independent in $R^n$

Is that true for y1 y2 and y3?

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Well, normally with linear indepence problems, I can set up a matrix and check to see if the row echelon form has free variables or not, or I can calculate the determinant of a square matrix. Here, I'm just given vectors x1 x2 and x3. What I did was set up a matrix like this with x1 as the first row, x2 as the second, and x3 as the third:

-1 1 0 = 0
0 -1 1 = 0
-1 0 1 = 0

And I end up with the following matrix

1 0 -1 0
0 1 -1 0
0 0 0 0

So I assume these vectors are linearly dependent (not independent) due to the free "variable" of x3?

 

[LCKurtz]

Or you might note y₃ = y₁ + y₂.

 

[DMOC]

Yes, that should be correct ... my bad. But would the longer method be valid? i.e. mathematically legal?

 

[LCKurtz]

Yes, that should be correct ... my bad. But would the longer method be valid? i.e. mathematically legal?

Yes.

 

[DMOC]

Thank you.