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    Linear algebra proofs

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    Please help with the following problems.

    1. Let u1 = (1,2,1,-1) and u2 = (2,4,2,0). Extend the linearly independent set {u1,u2} to obtain a basis for R4 (reals in 4 dimensions)

    2. Let U1,U2 be two subspaces of a finite dimensional vector space V such that U1+U2 = V. Prove that there is a subspace W of U1 such that W (+) U2 = V. [as in proof of dimension thm, extend to a basis of U1 n U2to obtain a basis of U1. consider the span of the vectors that have been added]
    express dim W in terms of dimensions of U1 and U1 n U2.
    ## [[ (+) represents + sign in a circle ]]

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    Solution Preview

    Please see the attached file.

    1) Consider the given vectors as row matrices.

    Write down the echlon matrix whose rows include the given vectors , namely

    1 2 1 -1 ( u1 )

    0 1 0 0

    2 ...

    Solution Summary

    This shows how to extend a linearly independent set to obtain a basis, and complete a proof regarding subspaces of finite dimensional vector space.