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Math Help - very important to me!

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    very important to me!

    Let F be a field, U and V vector spaces above field F and let T:V-->W be a linear transformation.
    Prove or disprove the next statements:

    1. If v_1, v_2, ... , v_n are vectors in vector space V and they are independent linear, then so T(v_1), ...T(v_n) independent linear.

    2. If v_1, v_2, ... , v_n are vectors in vector space V and they are dependent linear, then so T(v_1), ...T(v_n) dependent linear.

    I hope you understood my question, eve thou my broken English.

    I will appreciate a full answer!

    Thank you!
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  2. #2
    Senior Member Shanks's Avatar
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    the first statement is false. A couterexample is T=0 for all vectors in V.
    the second statement is true. Because a linear transformation preserve addition and multiplication by scalar, so if v_1,...v_n are linear dependent,that is, there exist nubmers a_1,...a_n (not all 0) in F such that the linear combination of v_1,...v_n is 0, thus the image of this linear combination under T is 0 in W, that is T(v_1), ...T(v_n) is dependent linear.
    To summarize, A linear transform T preserve the linear independency iff T is injective( T is invertable).
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