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Math Help - Linear Algebra Proofs regarding subspaces and spans

  1. #1
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    Linear Algebra Proofs regarding subspaces and spans

    1. Prove or give a counterexample to the following claim:
    Claim: Let V be a vector space over the field F and suppose that W1, W2 and W3
    are subspaces of V such that W1 + W3 = W2 + W3. Then W1 = W2.

    2. Consider the following subspaces of the vector space R^3
    over the field R of real numbers:
    subspace U1, which is the plane x + y + z = 0 and subspace U2, which is the yz-plane.
    a) Can R^3 be written as a sum of U1 and U2? Justify your answer.
    b) Can R^3 be written as a direct sum of U1 and U2? Justify your answer.
    Here x, y and z denote the usual Cartesian coordinates.


    3. Let V be a vector space over the field F and suppose (v1, v2, ... , vn) is
    a linearly independent set of vectors in V . Now suppose there exists w in V such
    that (v1 + w, v2 + w, ... , vn + w) is a linearly dependent set of vectors in V . Prove
    that w in span(v1, v2, ... , vn).

    Thank you.
    Last edited by zachoon; February 8th 2013 at 12:17 PM.
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  2. #2
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    Re: Linear Algebra Proofs regarding subspaces and spans

    I would really like to see how you would at least attempt these. For example, to show that W1= W2, you must show "if vector v is in W1 then v is in W2" and "if vector v is in W2 then it is in W1". If vector v is in W1 then, for any vector, u, in W3, v+ u is in W1+ W3. Because W1+ W3= W2+ W3, v+u is in W3. Therefore, ...
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  3. #3
    Senior Member jakncoke's Avatar
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    Re: Linear Algebra Proofs regarding subspaces and spans

    For
    1)
    Take W_1 = Span(\begin{bmatrix}1\\0\\0 \end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix})  , W_3 = Span(\begin{bmatrix}1\\0\\0 \end{bmatrix}), W_2 = Span(\begin{bmatrix}0\\1\\0 \end{bmatrix})

    Now it is true W_1 + W_3 = W_2 + W_3 but is W_1 = W_2 ?

    For 3)

    Consider what it means to be linearly dependent, the homogenrous eqn Ax = 0 has atleast one non trivial (atleast one coordinate or entry is non zero)

    so you have c_1(v_1+w) + ... + c_n(v_n+w) = 0 and  c_{i}(v_i+w) + ... + c_{k}(v_m+w) = 0 where i,..,k \in A (index set which contains the indicies of the non zero coordinates for the homogernous eqn). or basically  c_{i}v_i +... + c_{k}v_k + c_{i}w+...+c_{k}w = 0 or  c_{i}v_i +... + c_{k}v_k = c_{i}w+...+c_{k}w = (c_{i}+..+c_{k})w Divide both sides by (c_{i}+..+c_{k}) Why Can we Say with a gurantee that (c_{i}+..+c_{k}) wont equal zero? Think about that one.

    For
    2)
    a)yes
    b)no

    You have to justify it yourself, write out here and i can guide you.
    Last edited by jakncoke; February 8th 2013 at 02:01 PM.
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