# 8/29/2008

## Why study linear algebra?

• Systems of linear equations have interesting applications in many fields
• Historical problems are interesting and challenging
• Visualization and geometry, especially with rotations, reflections, and projections, are fascinating
• Provides a great introduction to abstract mathematics

## An Introduction to Linear Systems

• Systems with a unique solution
• Systems with infinitely many solutions
• Systems with no solutions

## Matrices

• A matrix is a rectangular array of numbers.
• See P:\data\math\hartlaub\linear algebra\MapleCommands.mw

## Vectors

• A matrix with only one column is called a column vector.
• A matrix with only one row is called a row vector.
• See P:\data\math\hartlaub\linear algebra\Vectors.mw

## Reduced row-echelon form

A matrix is in reduced row-echolon form if it satisfies all of the following conditions:

1. If a row has nonzero entries, then the first row entry is 1, called the leading 1 in this row.
2. If a column contains a leading 1, then all other entries in that column are zero.
3. If a row contains a leading 1, then each row above contains a leading 1 further to the left

## Gauss-Jordan Elimination

A method to solve systems of linear equations.

Begin by writing the augmented matrix of the system. Place a cursor in the top entry of the first nonzero column of this matrix.

1. If the cursor entry is zero, swap the cursor row with some row below to make the cursor entry nonzero.
2. Divide the cursor row by the cursor entry.
3. Eliminate all other entries in the cursor column, by subtracting suitable multiples of the cursor row from the other rows.
4. Move the cursor one row down and one column to the right. If the new cursor entry and all entries below are zero, move the cursor to the next column (remaining in the same row). Repeat the last step if necessary.

Return to Step 1. The process ends when we run out of rows or columns. Then, the matrix is in reduced row-echelon from (rref).

See P:\data\math\hartlaub\linear algebra\GaussJordanElimination.mw

## Exercises for next class meeting

• Section 1.1 (page 5) - 6, 12, 27