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Math Help - I need help on this Dimension theory HW (linear Alg)

  1. #1
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    I need help on this Dimension theory HW (linear Alg)

    Let V1,V2,.....Vk,V be vectors in a vector space V. and define W1 to be span(v1,V2,.....Vk) and W2 to be the Span(V1,V2.....Vk,V)

    1) Find the necessary conditions on V such that dim(W1)=dim(W2)

    2) State and prove a relationship involving dim(W1) and dim(W2), in the case where they dont equal each other.
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  2. #2
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    Hello,

    I can give you the solutions, and let you do the proof ! That's a good exercise.
    (and this is also because I don't know how to do it )

    Quote Originally Posted by JCIR View Post
    Let V1,V2,.....Vk,V be vectors in a vector space V. and define W1 to be span(v1,V2,.....Vk) and W2 to be the Span(V1,V2.....Vk,V)

    1) Find the necessary conditions on V such that dim(W1)=dim(W2)
    V is a linear combination of V1,V2,.....,Vk

    2) State and prove a relationship involving dim(W1) and dim(W2), in the case where they dont equal each other.
    Hmm dim(W1)<dim(W2)
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    Quote Originally Posted by JCIR View Post
    Let V1,V2,.....Vk,V be vectors in a vector space V. and define W1 to be span(v1,V2,.....Vk) and W2 to be the Span(V1,V2.....Vk,V)

    1) Find the necessary conditions on V such that dim(W1)=dim(W2)

    2) State and prove a relationship involving dim(W1) and dim(W2), in the case where they dont equal each other.
    Note W_1\subseteq W_2 thus \mbox{dim}(W_1) = \mbox{dim}(W_2) if and only if W_1 = W_2 if and only if V \in \text{spam}(V_1,...,V_k).

    If they do not equal to eachother then we know that the dimension must be stricly less, so, \text{dim}(W_1) < \text{dim}(W_2).
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  4. #4
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    Quote Originally Posted by ThePerfectHacker View Post
    if and only if V \in \text{spa{\color{red}m}}(V_1,...,V_k)
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