row equivalence and column equivalence

[v9260019]

hello all ^^
i have some question about row and column equivalence:
a matrix A by applying row operations we obtain a new matrix B,then A is row equivalence to B.
a matrix C by applying column operations we obtain a new matrix D, then C is column equivalence to D

1.What is the relationship between row equivalence and column equivalence??
2.What is the "meaning" of "equivalence"??

thanks a lot

 

[knmaheshy2k]

hi,
i dont think there is any relationship between row and column equivalence. Equivalence means we can get back one from other by doing some mathematical manupulations...
happy learning...

 

[steve10]

There are a couple of terms you need to give your definitions, operation and equivalence. However, according to your statement, "a matrix A by applying row operations we obtain a new matrix B,then A is row equivalence to B ...", you probably only need to define your "operation", so that we may come up something with or "guess" something from it.

Since we don't have that information, here is my loud guess.
A linear system can be written in a matrix form:
AX=B
where A is a matrix, and X and B are two column vectors where B is known while X needs to be found. Here are some basic facts about linear systems:
(1) The system will not change if the both sides of any equation is multiplied by a non-zero constant;
(2) The system will not change if any two equations switch positions;
(3) The system will not change if one equation is multiplied by any number and then added to any other equation.

In other words, the three "operations" produce a completely equivalent linear system. Therefore, we conclude that the "operations" also produce an equivalent matrix owing to the expression:
AX=B.

The corresponding operations to the matrix are:
(1) Any row is multiplied by a non-zero constant;
(2) Any two rows switch positions;
(3) Any row is multiplied by a number and then added to any other row.

Notice that this equivalence makes sense only in this special situation (in accordance with solving linear system). For example, the eigenvalues will vary if you multiply a row by a constant.

As for column equivalence, you may try to write a linear system as
XA=B
where X and B are row vectors.