Geometric Transformations

Let be defined by where is a nonzero vector. Show that is linear, and find the standard matrix of is the plane has scalar equation . (see projection)

We first need to show that is linear using the fact that projections are linear. For , and ,

We will skip some calculations, and we find that:

where each was calculated separately.

is the reflection in a hyperplane through the origin with normal vector in .

Let be a counterclockwise rotation about the origin by angle

we have that $\vec{x} = \bmatrix{r\cos(\phi) \\ r\sin(\phi)}$ (see Trig Functions, Polar Coordinates).

We will use the compound angle identities to get the standard matrix of this transformation:

(see matrix-vector product)

Let be a counterclockwise rotation about the origin by an angle . The standard matrix for this transformation is:

(see Trig Functions)

For , a positive real number:

Stretch in the direction:

Stretch in the direction:

When , the transformation becomes a compression.

If you try applying the matrix-vector product to a 2d vector, what happens?

So the component of the vector is stretched.

For , a positive real number:

When , the transformation becomes a contraction.

This is basically a stretch, but equally in all direction. It's just a resize then. Using the same intuition as stretches and compressions, you can see how all components of the vector will be changed.

Shears? Like in Minecraft???

For :

Shear in the direction:

Shear in the direction:

If we try and apply this transformation onto a vector, we get: