Equivalence relation demonstration - confirmation needed, please.

[mathstudent79]

Homework Statement

'Show that row equivalence is an equivalence relation'.

Homework Equations

The definition for 'row equivalence' given in the text is,

'two augmented matrices corresponding to linear systems that actually have solutions, are said to be (row) equivalent if they have the same solutions'.

The Attempt at a Solution

To show an equivalence relation, one must show reflexivity, symmetry and transitivity.

reflexivity:

any augmented matrix x clearly has the same solution set as itself.

symmetry:

Suppose that augmented matrices x and y are row equivalent. Then x and y have the same solution, by our definition. If x and y have the same solution, then y and x have the same solution, so y is also equivalent to x.

transitivity:

Suppose that we have augmented matrices x, y and z. And suppose that x is row equivalent to y, and y is row equivalent to z. Then, by definition, x and y have the same solution, and y and z have the same solution. Since x and y and z have the same solution, then x and z have the same solution. Therefore, x is row equivalent to z.

So my question is: did I do this correctly?

(NB: After doing this, I found a few proofs for this statement, which were rather elegant, but which involved inverse notation, or other notation which has not yet been presented in the book I'm going through. All that has been presented is row operations, Gaussian elim and the rather intuitive definition of row equivalence quoted above. I'm trying to 'show' the equivalence relation in terms of what I've been given so far. Thanks for any help!).

 

[vela]

Looks fine.

I suppose if you wanted to, you could be more formal about what it means to say that two systems have the same solutions.

 

[mathstudent79]

vela,

thank you VERY MUCH.

I will re-write it,making the adjustment that you suggest.

thanks again!

have a great day.