# Linear algebra proofs

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 ]]

https://brainmass.com/math/linear-algebra/linear-algebra-proofs-68001

#### 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.