## 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.

## 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.

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

## Row operations (with example)

### January 4, 2008

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

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**1. Swapping two rows:**

Here is a swap of rows 2 and 3. I’ll denote it by .

## Vector Space

### January 4, 2008

**What is Vector Space?**

**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.

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**Formal Definition**

**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

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Useful link for * Matrix Row Operations*: