If w1 w2 and w3 are independent vectors show that the differ

If w_1, w_2, and w_3 are independent vectors, show that the differences v_1 = w_2 - w_3, v_2 = w_1 - w_3, and v_3 = w_1 - w_2 are dependent. Do this by finding a combination of the v s that gives 0.

Solution

v1 and v2 and v3 are linearly dependent.

v3 = v2 - v1

w1 - w2 = (w1 - w3) - (w2 - w3)

= w1 - w2 - w3 +w3

= w1 - w2 = v3

So, v3 = v2 - v1

v3 - v2 + v1 =0   .

we have expressed one vector as a linear combination of other two