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  1. #1
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    Linear Algebra help!

    Hi I'm new here and I have a bit of trouble understanding a problem that was given to us:

    Let V and W be finite-dimensional vector spaces and let T:V->W be an isomorphism. Let Vo be a subspace of V. Prove that 1) T(Vo) is a subspace of W, and 2) dim (Vo)=dim(T(Vo))

    I would appreciate any help. thank you
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by x3non25 View Post
    Hi I'm new here and I have a bit of trouble understanding a problem that was given to us:

    Let V and W be finite-dimensional vector spaces and let T:V->W be an isomorphism. Let Vo be a subspace of V. Prove that 1) T(Vo) is a subspace of W, and 2) dim (Vo)=dim(T(Vo))

    I would appreciate any help. thank you

    the first one is easy..
    i)
    clearly, T(V_0) \subseteq W by taking an arbitrary element of V_0 \subseteq V and apply the mapping.

    ii)
    suppose u,w \in T(V_0), then there exist v_1, v_2 \in V_0 such that u=T(v_1) and w=T(v_2).
    now, u+w = T(v_1)+T(v_2) = T(v_1 + v_2) since T is an isomorphism. and since v_1, v_2 \in V_0, so does v_1 + v_2. thus u+v = T(v_1 + v_2) \in T(V_0)

    iii)
    Let a \in F and v \in T(V_0). Then \exists v_0 \in V_0 such that v = T(v_0)
    now, av = aT(v_0) = T(av_0) and just like the reasoning above, av = T(av_0) \in T(V_0).

    for the second one..
    i)
    let dim (V_0) = n. take \left\{ v_1, v_2, ..., v_n \right\} be a basis for V_0. Let w \in T(V_0). then, there exists a v \in V_0 such that w = T(v), say v = a_1v_1 + a_2v_2 + ... + a_nv_n..

    w = T(v) = T(a_1v_1 + a_2v_2 + ... + a_nv_n) = T(a_1v_1) + T(a_2v_2) + ... T(a_nv_n)
     =a_1T(v_1) + a_2T(v_2) + ... + a_nT(v_n)

    thus \left\{ T(v_1), T(v_2), ..., T(v_n)\right\} is a spanning set for T(V_0)

    ii)
    suppose a_1T(v_1) + a_2T(v_2) + ... + a_nT(v_n) = 0 (the zero vector for T(V_0)).
    then,
    a_1T(v_1) + a_2T(v_2) + ... + a_nT(v_n) = T(a_1v_1 + a_2v_2 + ... + a_nv_n) = 0
    \Longleftrightarrow a_1v_1 + a_2v_2 + ... + a_nv_n = 0 (the zero vector for V_0).. since \left\{ v_1, v_2, ..., v_n \right\} be a basis for V_0,
    a_1v_1 + a_2v_2 + ... + a_nv_n = 0 \Longleftrightarrow a_i = 0 \, \, \forall \, i = 1,...,n..

    this implies that \left\{ T(v_1), T(v_2), ..., T(v_n)\right\} is a basis for T(V_0). Therefore dim T(V_0) = n.. QED
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