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jackson6612

Member

Registered: 2006-07-19

Posts: 5

Matrices

Hi

Please keep in mind while answering this post that I'm first year college student so please keep your answers simple. Thank you

Elementary Row and Column operations:

Usually a given system of linear equations is reduced to a simple equivalent system by applying in turn a finite number of elementary operations which are stated below:

1: Interchanging two equations

2: Multiplying an equation by a non-zero number.

3: Adding a multiple of one equation to another equation.

Corresponding to these three elementary operations, the following elementary row operations are applied to matrices to obtain equivalent matrices ( these equivalent matrices still represent the same system of linear equations ). The elementary row operations are:

1: Interchanging two rows

2: Multiplying a row by a non-zero number.

3: Adding a multiple of one row to another row.

There are also elementary column operations which are given below:

1: Interchanging two columns

2: Multiplying a column by a non-zero number.

3: Adding a multiple of one column to another column.

Consider the following system:

x + y + 2z = 1
2x - y +8z = 12
3x + 5y + 4z = -3

which can be written in matrix form as AX=B, where

A = [1,1,2; 2,-1,8; 3,5,4]
X = [x; y; z]
B = [1; 12; -3]

A is called the matrix of coefficients.

Appending a column of constants which is B on the left of A, we get the augmented matrix of given system.

Augmented Matrix = [1,1,2:1; 2,-1,8:12; 3,5,4:-3]

If you see this augmented matrix carefully, it's the the same system of linear equations, each row represents a equation, except that we have omitted the variables and we have colon instead of 'equal to' sign.

Now we can apply the elementary row operations on augmented matrix.

But to apply elementary column operations we have to write this augmented matrix differently, like this:

[1,2,3; 1,-1,5; 2:1,8:12,4:-3]

This augmented matrix still represents the same system of linear equations except that now each column represents a equation.

QUESTIONS:

1: Am I right that while applying elementary column operations we write augmented matrix in such a way that each column represents equation?

2: Don't these row and column operations effect determinant?

Row Echelon Form of Matrix:

We can get a special form of matrix A which is called row echelon form by applying row operations and that row echelon form of matrix A should meet the the following conditions.

1: In each successive non-zero row, the number of zeros before the leading entry ( first non-zero entry in a certain row is known as leading entry ) is greater than the number of zeros in the preceding row.

2: Every non-zero row precedes every zero row ( if any ).

3: The non-zero entry ( leading entry ) in each row is 1.

4 : A is said to be in reduced row echelon form if it is in row echelon form and if the first non-zero entry ( leading entry ) in Row(i) lies in Column(j), then all the other entries of Column(j) are zero.

This fourth condition is used for reduced row echelon form of matrix.

Row echelon form of the this augmented matrix,

Augmented Matrix = [1,1,2:1; 2,-1,8:12; 3,5,4:-3]

is [1,0,0:1; 0,1,0:-2; 0,0,1:1]

QUESTIONS:

1: What are conditions for column row echelon form of matrix?

2: What will be column echelon form this augmented matrix,
Augmented Matrix = [1,2,3; 1,-1,5; 2:1,8:12,4:-3]?

3: Does the concept of echelon form of matrix only applies to augmented matrices?

Row Rank of Matrix:

Let C be a non-zero matrix. If r is the number of non-zero rows when it is reduced to the reduced echelon form, then r is called the row rank of the matrix A.

Reduced echelon form of matrix C,

C = [1,-1,2,-3; 2,0,7,-7; 3,1,12,-11]

is [1,0,7/2,-7/2; 0,1,3/2,-1/2; 0,0,0,0].

So rank of C is 2.

I also tried to find the column rank of C. For that I wrote C differently so that I can apply column operations.

C = [1,2,3; -1,0,1; 2,7,12; -3,-7,-11]

I don't think that C is in column reduced echelon form. But as you can see I can't apply column operation on C any further.

Suspected column echelon form of C is:
[1,0,0; 0,1,0; 7/2,3/2,0; -7/2,-1/2,0]

So I think row rank is also 2.

QUESTIONS:

1: Am I right about the column rank of C?

2: Are row and column ranks of certain matrix always have to be same?

Linearly Independent Rows and Columns:

I think in this matrix [1, 0, 0; 0, 1, 0; 0, 0, 1] all rows are independent. While in matrix [2,-1,1; 1,0,1; 3,-1,2], the third row is not independent because it is the sum of the first two rows.

QUESTIONS:

1: But what's this 'LINEAR' independence?

2: Shall we say that in matrix [2,-1,1; 1,0,1; 3,-1,2], only first two rows are independent?

Please give me some example about column independence.

Sincerely,
Vijay

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Matrices

you mean
4
=
x
-y
+3z
?

Sure this is also an equation.

But since matrix product has its direction and rules, you should rewrite the equation like
x'A'=c' where ' means transpose. The equation is just equavalent to Ax=c

Good thinking! You've been very advanturous. Your question is what you goona learn maybe 1 or 2 months later.

About determinants, they are very delicate beings, transpose them wouldn't change the value. I can give you a simple example though the entire proof is tedious.

|2 1 |=-2-3   
|3 -1|

|2 3|=-2-3
|3 -1|

for any matrix A
the rank of row1(A) row2(A) ... rowm(A)
is equal to
the rank of col1(A) col2(A) ... coln(A)
which means row rank and column rank are the same.

Still, I cannot give you aproof since it's not only hard but also long.

Linear independence
If you cannot express a vector ([0 1 0] for example) by coefficiently adding a group of some vectors anyway, you get it independent from the group of given vectors.
coefficiently adding-
[1 0 0]+[0 0 1] is adding, also coefficiently adding
2[1 0 0]+(-1)[0 0 1] is coefficiently adding
so do 0[1 0 0]+0[0 0 1]
apparently [0 0 0] is never independent.

if you get a group of vectors independent, you mean none of them can be expressed as the coefficient sum of others.

By the way, coefficient sum and coefficiently adding is first used by myself as far as I know, and I guess the reason is that I'm just too lazy to write the whole expression. Hope you could understand it!

for more information, recommand you get a detailed Linear Matrix textbook.

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