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## Homework Statement

Suppose that pairs of vectors ##v_1, v_2## and ##v_1, v_3## and ##v_2, v_3## are linearly independent. Must ##v_1, v_2, v_3## be linearly independent?

## Homework Equations

So this means ##v_1 \neq av_2##, ##v_1 \neq bv_3## and ##v_2 \neq cv_3##

## The Attempt at a Solution

For the sake of intuition, I started playing around with vectors in ##\mathbb {R}^3##. And I came up with values for ##v_1, v_2##, and ##v_3## that are independent as pairs but linearly dependent when together:

(1,2,1)

(0,1,1)

(1,1,0)

So I've answered the question by existence but this doesn't feel very satisfying, especially since I only shown this for ##\mathbb {R}^3## and not shown it in general. Do you have suggestions for a better proof?