# Vectors and Matrices

Last revision: January 11, 2017In this section we are going to cover some linear algebra
stuff. `Octave`

is especially well designed for those purposes.

Vectors are the most basic objects in `Octave`

. You already
know how to define some: `V=1:5`

yields `V=(1,2,3,4,5)`

. You can also
say:

But that’s a *row vector*. What if you want a *column* vector? There
are two ways:

which means *transpose* (i.e.: convert rows into columns), or either
you can do:

Either way it will work. You can *cut* vectors easily:

Matrices are not difficult to define either:

Of course, you can multiply a matrix with a column vector:

Solving a linear system is very easy. If your system is $A\vec x=\vec
b$, with $A$ a matrix and $\vec b$ a vector, then you can formally
write `x = A \ b`

, and that will solve the system by gaussian
reduction. An example, let’s solve the system:

So we write the matrix $A$ and the right hand side vector $\vec b$:

We can check easily that the solution is right:

as it should.

The special matrices are easy to find. For example, the identity, $I$, is called “eye” (yes, it’s someone’s idea of a joke):

Or the zero matrix:

## Exercises

- Solve the following linear system of equations:

Check the solution!

- Find the 10-th power of the following matrix