CHAPTER I: LINEAR ALGEBRA - OCW UPM

3. Let us check that G 3 = {(1, 1, 1), (2, 1, 1), (0, 0, 1), (3, 0, 0)} is a generating
system of R 3 . Given (x 1 , x 2 , x 3 ) ∈ R 3 . We wonder if there exist
λ 1 , λ 2 , λ 3 , λ 4 ∈ R such that
(x 1 , x 2 , x 3 ) = λ 1 (1, 1, 1) + λ 2 (2, 1, 1) + λ 3 (0, 0, 1) + λ 4 (3, 0, 0)
equivalently if the following system with unknown variables λ 1 , λ 2 , λ 3 y
λ 4 has a solution ⎧
⎨ λ 1 + 2λ 2 + 3λ 4 = x 1
λ
⎩ 1 + λ 2 = x 2
λ 1 + λ 2 + λ 3 = x 3 .
The coefficient matrix has rank 3 so Rouche’s Theorem allows to say
that for every (x 1 , x 2 , x 3 ) ∈ R 3 the system has infinitely many solutions.