Linear Algebra, 2020a

250 Chapter Three. Maps Between Spaces
will, when it acts from the left, perform the combination operation −2ρ 2 + ρ 3 .
⎛
⎜
1 0 0
⎞ ⎛
⎟ ⎜
1 0 2 0
⎞ ⎛
⎟ ⎜
1 0 2 0
⎞
⎟
⎝0 1 0⎠
⎝0 1 3 −3⎠ = ⎝0 1 3 −3⎠
0 −2 1 0 2 1 1 0 0 −5 7
3.19 Definition The elementary reduction matrices (or just elementary matrices)
result from applying a single Gaussian operation to an identity matrix.
(1) I
kρ i
−→ Mi (k) for k ≠ 0
(2) I
ρ i ↔ρ j
−→
Pi,j for i ≠ j
(3) I
kρ i +ρ j
−→
Ci,j (k) for i ≠ j
3.20 Lemma Matrix multiplication can do Gaussian reduction.
(1) If H kρ i
−→ G then M i (k)H = G.
(2) If H ρ i↔ρ j
−→ G then Pi,j H = G.
(3) If H kρ i+ρ j
−→ G then Ci,j (k)H = G.
Proof Clear.
QED
3.21 Example This is the first system, from the first chapter, on which we
performed Gauss’s Method.
3x 3 = 9
x 1 + 5x 2 − 2x 3 = 2
(1/3)x 1 + 2x 2 = 3
We can reduce it with matrix multiplication. Swap the first and third rows,
⎛
⎜
0 0 1
⎞ ⎛
⎞ ⎛
⎞
0 0 3 9 1/3 2 0 3
⎟ ⎜
⎟ ⎜
⎟
⎝0 1 0⎠
⎝ 1 5 −2 2⎠ = ⎝ 1 5 −2 2⎠
1 0 0 1/3 2 0 3 0 0 3 9
triple the first row,
⎛
⎜
3 0 0
⎞ ⎛
⎞ ⎛
⎞
1/3 2 0 3 1 6 0 9
⎟ ⎜
⎟ ⎜
⎟
⎝0 1 0⎠
⎝ 1 5 −2 2⎠ = ⎝1 5 −2 2⎠
0 0 1 0 0 3 9 0 0 3 9