# Math 790.03: Linear Algebra

This is the webpage for the Fall 2018 manifestation of Dr. Fitzpatrick's section of Math 790-92.03 at Duke University.

Beezer's *A First Course in Linear Algebra* is a nice freely available
reference.

## SageMath

SageMath is an open source computer algebra system written python.

## Useful Commands

Use the syntax `A = matrix([(1/2,-3,5),(-4/5,2,11),(6,-4,9)])`

to define the matrix
\[
A
=
\left[\begin{array}{rrr}
\frac{1}{2} & -3 & 5 \\
-\frac{4}{5} & 2 & 11 \\
6 & -4 & 9
\end{array}\right]
\]
Avoid using decimal notation! Input `1/2`

instead of `.5`

.

- The command
`A.rref()`

computes the reduced row-echelon form of`A`

. - The command
`A.det()`

computes the determinant of`A`

, provided that`A`

is square. - The command
`A.rank()`

computes the rank of`A`

. - If
`A`

and`B`

are \(m\times n\) matrices, then the command`A+B`

computes the sum of`A`

and`B`

. - If
`c`

is a scalar, then the command`c*A`

computes the scalar product of`c`

and`A`

. - If
`A`

is \(\ell\times m\) and`B`

is \(m\times n\), then`A*B`

computes the product of`A`

and`B`

. - The command
`A.transpose()`

computes the transpose of`A`

. - The command
`A.is_symmetric()`

tests if`A`

is symmetric.

## Reduced Row-Echelon Form

The following code computes the reduced row-echelon form of a given matrix \(A\).

## Symbolic Row-Reductions

The following code produces the reduced form of an augmented matrix with symbolic variables in the augmented column.

## Reduced Row-Echelon form Practice!

Enter the number of rows and the number of columns of the matrix.

## The Four Fundamental Subspaces

Enter the matrix.

To practice finding bases, you can generate a random \(m\times n\) matrix with
the command `A = random_matrix(ZZ, m, n)`

.

A random \(m\times n\) matrix with rank \(r\) whose reduced row-echelon form has
integer entries can be generated with the command ```
A = random_matrix(ZZ, m, n,
algorithm='echelonizable', rank=r)
```

.