Page 40 - 35Linear Algebra

40 Systems of Linear Equations

2.1.2 Equivalence and the Act of Solving

We now introduce the symbol ∼ which is called “tilde” but should be read as
“is (row) equivalent to” because at each step the augmented matrix changes
by an operation on its rows but its solutions do not. For example, we found
above that

1 1 27 1 0 9 1 0 9
∼ ∼ .
2 −1 0 2 −1 0 0 1 18
The last of these augmented matrices is our favorite!

Equivalence Example

Setting up a string of equivalences like this is a means of solving a system
of linear equations. This is the main idea of section 2.1.3. This next example
hints at the main trick:

Example 12 (Using Gaussian elimination to solve a system of linear equations)

x + y = 5 1 1 5 1 1 5 1 0 2 x + 0 = 2
⇔ ∼ ∼ ⇔
x + 2y = 8 1 2 8 0 1 3 0 1 3 0 + y = 3
Note that in going from the first to second augmented matrix, we used the top left 1
to make the bottom left entry zero. For this reason we call the top left entry a pivot.
Similarly, to get from the second to third augmented matrix, the bottom right entry
(before the divide) was used to make the top right one vanish; so the bottom right
entry is also called a pivot.

This name pivot is used to indicate the matrix entry used to “zero out”
the other entries in its column; the pivot is the number used to eliminate
another number in its column.

2.1.3 Reduced Row Echelon Form

For a system of two linear equations, the goal of Gaussian elimination is to
convert the part of the augmented matrix left of the dividing line into the
matrix

1 0
I = ,
0 1

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