Linear Algebra, 2020a

56 Chapter One. Linear Systems
III.2
The Linear Combination Lemma
We will close this chapter by proving that every matrix is row equivalent to one
and only one reduced echelon form matrix. The ideas here will reappear, and be
further developed, in the next chapter.
The crucial observation concerns how row operations act to transform one
matrix into another: the new rows are linear combinations of the old.
2.1 Example Consider this Gauss-Jordan reduction.
(
2 1 0
1 3 5
)
−→
2 1 0
0 5/2
)
5
( )
1 1/2 0
0 1
(
2
)
1 0 −1
−→
0 1 2
−(1/2)ρ 1 +ρ 2
(
(1/2)ρ 1
−→
(2/5)ρ 2
−(1/2)ρ 2 +ρ 1
Denoting those matrices A → D → G → B and writing the rows of A as α 1 and
α 2 , etc., we have this.
( )
(
)
α 1 −(1/2)ρ 1 +ρ 2 δ 1 = α 1
−→
α 2 δ 2 =−(1/2)α 1 + α 2
(
(1/2)ρ 1
−→
(2/5)ρ 2
(
−(1/2)ρ 2 +ρ 1
−→
)
γ 1 =(1/2)α 1
γ 2 =−(1/5)α 1 +(2/5)α 2
)
β 1 =(3/5)α 1 −(1/5)α 2
β 2 =−(1/5)α 1 +(2/5)α 2
2.2 Example The fact that Gaussian operations combine rows linearly also holds
if there is a row swap. With this A, D, G, and B
( ) ( ) ( ) ( )
0 2 ρ 1 ↔ρ 2 1 1 (1/2)ρ 2 1 1 −ρ 2 +ρ 1 1 0
−→ −→ −→
1 1
0 2
0 1
0 1
we get these linear relationships.
( ) ( )
⃗α 1 ρ 1 ↔ρ 2 ⃗ δ 1 = ⃗α 2
−→
⃗α 2
⃗δ 2 = ⃗α 1
(
(1/2)ρ 2
−→
)
⃗γ 1 = ⃗α 2
⃗γ 2 =(1/2)⃗α 1
( )
−ρ 2 +ρ 1 β ⃗ 1 =(−1/2)⃗α 1 + 1 · ⃗α 2
−→
⃗β 2 =(1/2)⃗α 1
In summary, Gauss’s Method systematically finds a suitable sequence of
linear combinations of the rows.