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Math Help - Linear Independence

  1. #1
    Senior Member I-Think's Avatar
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    Linear Independence

    Let [v_1, v_2, v_3] be a linearly independent subset of a vector space over C. Determine
    all scalars \delta  with which [v_1-\delta{v_2}, \delta{v_1} + v2, v_1 + v_2 + v_3] is linearly independent.

    Attempt so far

    If [v_1, v_2, v_3] is linearly independent, then for a_i\in{C}
    a_1v_1+a_2v_2+a_3v_3\neq{0}<br />
    And consider
    a_1v_1+a_2v_2+a_3v_3+v_1(\delta{a_2}+a_3)+v_2(-\delta{a_1}+a_3)\neq{0}<br />
    How do I prove linear independence from here?
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by I-Think View Post
    Let [v_1, v_2, v_3] be a linearly independent subset of a vector space over C. Determine
    all scalars \delta with which [v_1-\delta{v_2}, \delta{v_1} + v2, v_1 + v_2 + v_3] is linearly independent.

    Attempt so far

    If [v_1, v_2, v_3] is linearly independent, then for a_i\in{C}
    a_1v_1+a_2v_2+a_3v_3\neq{0}<br />
    And consider
    a_1v_1+a_2v_2+a_3v_3+v_1(\delta{a_2}+a_3)+v_2(-\delta{a_1}+a_3)\neq{0}<br />
    How do I prove linear independence from here?
    Hint: consider the determinant of the matrix

    \begin{vmatrix}1 & \delta & 1 \\ -\delta & 1 & 1\\ 0 & 0 & 1 \end{vmatrix}=1+\delta^2

    Why does this help?
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