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Math Help - Another linear dependence problem

  1. #1
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    Another linear dependence problem

    if \{u,v,w,\} is a linearly independent set of vectors, then S = \{u+v-2w,u-v-w,u+w\} is also linearly independent.

    True or false?

    thanks
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  2. #2
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    Quote Originally Posted by Bucephalus View Post
    if \{u,v,w,\} is a linearly independent set of vectors, then S = \{u+v-2w,u-v-w,u+w\} is also linearly independent.

    True or false?

    thanks
    Form the determinant of those coefficients and show it is non-zero.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Form the determinant of those coefficients and show it is non-zero.
    At first I set up the matrix with each vector in a row like this:

    <br />
\left(<br />
\begin{matrix}<br />
u & v & -2w\\<br />
u & -v & -w\\<br />
u & 0 & w\\<br />
\end{matrix}<br />
\right)<br />

    But then I couldn't see how this would work. Then I realised that you meant the following.

    <br />
\left(<br />
\begin{matrix}<br />
u & u & u\\<br />
v  & -v & 0\\<br />
-2w & -w & w\\<br />
\end{matrix}<br />
\right)<br />

    So I formed the determinant of this matrix and got a non-zero result.
    Thanks for your help.
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