The standard basis,

$e_1$ and

$e_2$, are unit vectors in

$\Bbb{R}^2$ such that:

If

are the transformed vectors, then the standard matrix is

Why does that work? Watch the intro video

The standard basis,

$e_1$,

$e_2$, and

$e_3$ are unit vectors in

$\Bbb{R}^3$ such that:

If are the transformed vectors, then the standard matrix is

To find the standard basis in $\Bbb{R}^2$ geometrically in a graph we:

1. Draw the standard basis $e_1$ and $e_2$ in the $x_1 x_2$ plane

2. Draw the transformed vectors using the information given

3. Identify the transformed vectors $T(e_1)$, and $T(e_2)$.

4. Combine them to get the standard matrix

Here are the many types of transformations you may see in this section:

__Reflections__

__Vertical/Horizontal Expansions and contractions__

__Vertical/Horizontal Shears__

__Projections__

__Circle Rotation__