Row-reducing Algorithms

January 4, 2008

Row reduction is the process of using row operations to transform a matrix into a row reduced echelon matrix. As the algorithm proceeds, you move in stairstep fashion through different positions in the matrix. In the description below, when I say that the current position is (i, j) , I mean that your current location is in row i and column j. The current position refers to a location, not the element at that location. The current row means the row of the matrix containing the current position.

Read the rest of this entry »

Row reduction

January 4, 2008

What is row reduction?

Row reduction (or Gauss-Jordan elimination) is the process of using row operations to reduce a matrix to row reduced echelon form. This procedure is used to solve systems of linear equations, invert matrices, compute determinants, and do many other things.

——————————-

A matrix is in row reduced echelon form if the following conditions are satisfied:

Read the rest of this entry »

There are three kinds of row operations. Actually, there is some redundancy here – you can get away with two of them.

1. Swapping two rows:

Here is a swap of rows 2 and 3. I’ll denote it by $r_2 \leftrightarrow r_3$.

row-switching

. Read the rest of this entry »

Vector Space

January 4, 2008

What is Vector Space?

A vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are defined and satisfy certain natural axioms.

——————

Formal Definition

Let F be a field (such as the real numbers or complex numbers), whose elements will be called scalars. A vector space over the field F is a set V together with two binary operations, Read the rest of this entry »

Matrix Row Operations

January 4, 2008

For matrices, there are three basic row operations. They are:

* Row switching: Interchange rows

* Row multiplication:  Multiply a row by a non-zero constant, changing that row

* Row addition: Use a pivot row to change a target row by multiplying the pivot row by a constant and adding the resulting products to the target row. The pivot row is not changed

————

Useful link for Matrix Row Operations:

http://www.purplemath.com/modules/mtrxrows.htm