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    linearly independent

    Suppose that S={v1, v2, v3} is a linearly independent set of vectors in a vector space V. How do you show that T={w1, w2, w3} is also linearly independent , where w1=v1+v2+v3, w2=v2+v3, and w3=v3?
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    Quote Originally Posted by antman View Post
    Suppose that S={v1, v2, v3} is a linearly independent set of vectors in a vector space V. How do you show that T={w1, w2, w3} is also linearly independent , where w1=v1+v2+v3, w2=v2+v3, and w3=v3?
    Assume the contrary, and suppose that T is linearly dependent. Then there must be some c_1,\;c_2,\;c_3, not all zero, such that

    c_1w_1+c_2w_2+c_3w_3=0

    \Rightarrow c_1(v_1+v_2+v_3)+c_2(v_2+v_3)+c_3v_3=0

    \Rightarrow c_1v_1+c_1v_2+c_1v_3+c_2v_2+c_2v_3+c_3v_3=0

    \Rightarrow c_1v_1+(c_1+c_2)v_2+(c_1+c_2+c_3)v_3=0.

    Since v_1,\;v_2,\text{ and }v_3 are linearly independent, we have

    \left\{\begin{array}{ccccccc}<br />
c_1&&&&&=0\\<br />
c_1&+&c_2&&&=0\\<br />
c_1&+&c_2&+&c_3&=0<br />
\end{array}\right.

    \Rightarrow c_1=c_2=c_3=0. But this contradicts the requirement that c_1,\;c_2,\;c_3 are not all zero. Hence, our initial assumption is false, and T is linearly independent.
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