Page 369 - 35Linear Algebra

G.2 Systems of Linear Equations 369

Hints for Review Question 10

This question looks harder than it actually is:
Row equivalence of matrices is an example of an equivalence
relation. Recall that a relation ∼ on a set of objects U
is an equivalence relation if the following three properties
are satisfied:

• Reflexive: For any x ∈ U, we have x ∼ x.
• Symmetric: For any x, y ∈ U, if x ∼ y then y ∼ x.

• Transitive: For any x, y and z ∈ U, if x ∼ y and y ∼ z
then x ∼ z.

(For a more complete discussion of equivalence relations, see
Webwork Homework 0, Problem 4)
Show that row equivalence of augmented matrices is an equivalence
relation.
Firstly remember that an equivalence relation is just a more general ver-
sion of ‘‘equals’’. Here we defined row equivalence for augmented matrices
whose linear systems have solutions by the property that their solutions are
the same.
So this question is really about the word same. Lets do a silly example:
Lets replace the set of augmented matrices by the set of people who have hair.
We will call two people equivalent if they have the same hair color. There are
three properties to check:
• Reflexive: This just requires that you have the same hair color as
yourself so obviously holds.

• Symmetric: If the first person, Bob (say) has the same hair color as a
second person Betty(say), then Bob has the same hair color as Betty, so
this holds too.

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