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 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.
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 ...
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