Page 54 - 35Linear Algebra
P. 54
54 Systems of Linear Equations
Example 24 (Undoing A in Ax = b slowly, for A = M = ...)
0 1 1 x 7
2 0 0 y = 4
0 0 1 z 4
0 1 0 0 1 1 x 0 1 0 7
⇔ 1 0 0 2 0 0 = 1 0 0
y
4
0 0 1 0 0 1 z 0 0 1 4
2 0 0 x 4
⇔ 0 1 1 =
7
y
0 0 1 z 4
1 0 0 2 0 0 x 1 0 0 4
2 2
⇔ 0 1 0 0 1 1 = 0 1 0
7
y
0 0 1 0 0 1 z 0 0 1 4
1 0 0 x 2
⇔ 0 1 1 =
y
7
0 0 1 z 4
1 0 0 1 0 0 x 1 0 0 2
⇔ 0 1 −1 0 1 1 = 0 1 −1
y
7
0 0 1 0 0 1 z 0 0 1 4
1 0 0 x 2
⇔ 0 1 0 = .
3
y
0 0 1 z 4
This is another way of thinking about Gaussian elimination which feels more
like elementary algebra in the sense that you “do something to both sides of
an equation” until you have a solution.
2.3.2 Recording EROs in (M|I )
−1 −1
Just as we put together 3 2 = 6 −1 to get a single thing to apply to both
sides of 6x = 12 to undo 6, we should put together multiple EROs to get
a single thing that undoes our matrix. To do this, augment by the identity
matrix (not just a single column) and then perform Gaussian elimination.
There is no need to write the EROs as systems of equations or as matrices
while doing this.
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