Matrix Mappings

Matrices are not only used to solve systems of equations, but are also used to map vectors into another vector in space.

A function that performs this mapping is an input vector multiplied by some transformation matrix.

The domain of a mapping is its number of columns and it’s codomain is its number of rows.

The Image is the output vector of the mapping.

Matrix mapping must satisfy…

1. $f_A(\vec{x}+\vec{y})=f_A(\vec{x})+f_A(\vec{y})$
2. $f_A(t\vec{x})=t{f}_A(\vec{x})$

Linear Mapping

This is defined as a mapping from ${\mathbb{R}}^n\to {\mathbb{R}}^m$ if the following is satisfied

1. $L(\vec{x}+\vec{y})=L(\vec{x})+L(\vec{y})$
2. $L(t\vec{x})=tL(\vec{x})$ where we can combine these properties in one example when we test.

Standard Matrix of Linear Mapping

You can represent all linear mappings as matrix mapping by forming a transformation matrix by compiling the images of the basis vectors.

For example, for a domain in ${\mathbb{R}}^2$, the basis is…
$\left[\begin{array}{c}0\\ 1\end{array}\right]$ and $\left[\begin{array}{c}1\\ 0\end{array}\right]$. We can apply a linear mapping of $(2x_1+3x_2,x_1)$ on this basis to get $\left[\begin{array}{c}3\\ 0\end{array}\right]$ and $\left[\begin{array}{c}2\\ 1\end{array}\right]$ making matrix $\left[\begin{array}{cc}3& 2\\ 0& 1\end{array}\right]$ where we can multiply an input vector by this transformation matrix to get our image.

Composite Mapping

You can represent two linear mappings in one matrix called a composition of linear mapping.

You can also think of this as a cascading mapping. The idea is that for two mappings of…
$L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ and $M:{\mathbb{R}}^m\to {\mathbb{R}}^p$, where the codomain of the first mapping must be the domain of the second mapping, the composition is defined as $(M\circ L)(\vec{x}=M(L(\vec{x})))$.

We need the codomain and domains to match up since matrix multiplication only works if the inner dimensions are equivalent.

Geometrical Transformations are too large of a topic to be included here.