- #1

SetepenSeth

- 16

- 0

## Homework Statement

Let { u, v, w} be a set of vectors linearly independent on a vector space V

- Is { u-v, v-w, u-w} linearly dependent or independent?

## Homework Equations

[/B]

A set of vectors u, v, w are linearly independent if for the equation

au + bv + cw= 0 (where a, b, c are real scalars)

The only solution is the trivial one a = b = c = 0

## The Attempt at a Solution

I expressed the vector in base of their coefficients, changed them into column vectors, formed a matrix and solve the homogeneous system

(1)u - (1)v + (0)w

(0)u + (1)v - (1)w

(1)u + (0)v - (1)w

| 1 0 1 |

|-1 1 0 |

| 0 -1 -1 |

After doing (R2 + R1 on R2) > (R3 + R2 on R3) I get:

| 1 0 1|

| 0 1 1|

| 0 0 0|

Which has no pivot on the 3rd column, thus the system has multiple solutions, making the set linearly dependent, however my answer key says is linearly independent and I'm not sure what I may be doing wrong.