Row Reduction Question: Independent w1 or w2? Answer Revealed!

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transgalactic
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which one of w1 or w2 linear independent of the v groupr??
i got to the situation that each one of them could be independent and the other not
??
the answer in the book is that w2 is independent and the other is not
[tex] v_1=\begin{bmatrix}<br /> 1\\ <br /> 1\\ <br /> -2\\ <br /> 1<br /> \end{bmatrix}<br /> v_2=\begin{bmatrix}<br /> 3\\ <br /> 0\\ <br /> 4\\ <br /> -1<br /> \end{bmatrix}<br /> v_3=\begin{bmatrix}<br /> -1\\ <br /> 2\\ <br /> 5\\ <br /> 2<br /> \end{bmatrix}<br /> w_1=\begin{bmatrix}<br /> 8\\ <br /> -10\\ <br /> 18\\ <br /> -14<br /> \end{bmatrix}<br /> w_2=\begin{bmatrix}<br /> 3\\ <br /> 4\\ <br /> 1\\ <br /> 1<br /> \end{bmatrix}[/tex]
http://img147.imageshack.us/img147/8002/64728417.th.gif
 
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did solved it correctly?
 
You shouldn't need to ask whether you solved it correctly, since it's easy enough to check.

Your problem boils down to two questions:
  1. Does w1 belong to the span of v1, v2, and v3? In other words, are there constants c1, c2, and c3 such that w1 = c1*v1 + c2*v2 + c3*v3? If you get a solution for the constants, w1 is in Span({v1, v2, v3, w1}), and so cannot be independent of them. If you don't get a solution, w1 isn't in Span({v1, v2, v3, w1}), and so must be independent of them.
  2. Does w2 belong to the span of v1, v2, and v3? Same as above.