what to do for this one:

Given: 3 vectors,a,b,c, are linearly independent in a vector space V

Asked: Show that 3a, 2a-b,a+care also linearly independent

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- Oct 22nd 2009, 04:15 PM #1

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- Oct 22nd 2009, 04:29 PM #2

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Assume 3a, 2a-b, a+c are dependent. Then, there exist scalras $\displaystyle x_1,x_2,x_3$, not all zero, such that $\displaystyle x_1(3a) + x_2(2a-b) + x_3(a+c) = 0 \Rightarrow 3x_1a + 2x_2a -x_2b + x_3a + x_3c = 0 $ $\displaystyle \Rightarrow (3x_1+2x_2+x_3)a + (-x_2)b + (x_3)c = 0$

If $\displaystyle x_2=x_3=0$ then $\displaystyle x_1 \neq 0$. Therefore not all the coefficients are zero.

Let $\displaystyle \lambda_1 = 3x_1+2x_2+x_3$

$\displaystyle \lambda_2 = -x_2$

$\displaystyle \lambda_3 = x_3$

Then $\displaystyle \lambda_1a + \lambda_2b + \lambda_3c = 0$ but not all of the coefficients are zero, in contradiction to the fact that a,b,c are linearly independent... ==> the assumption can not hold, therefore 3a, 2a-b and a+c are linearly independent.

- Oct 22nd 2009, 04:51 PM #3

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