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Math Help - linearly dependent

  1. #1
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    linearly dependent

    let T:R^n -> R^m be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in R^n. Explain why the set {T(v1),(v2),(v2)} is linearly dependent.
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  2. #2
    MHF Contributor kalagota's Avatar
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    first, the latter set contains two elements which are equal..

    assuming my corrections below...
    Quote Originally Posted by ssl000 View Post
    let T:R^n -> R^m be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in R^n. Explain why the set {T(v1),T(v2),T(v3)} is linearly dependent.
    if \{v_1, v_2, v_3\} is linearly dependent, then WLOG, v_1 = r_2v_2+ r_3v_3 where r_2 and r_3 are scalars..

    T(v_1) = T(r_2v_2+ r_3v_3) = r_2T(v_2) + r_3T(v_3) since T is a linear transformation.
    Thus you were able to express, WLOG, T(v_1) as a linear combination of T(v_2) and T(v_3). Hence \{T(v1),T(v_2),T(v_3)\}
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