Linear Transformations
Overview
A linear transformation is a function from a vector space V to a vector space W
{% T:V \rightarrow W %}
such that
- {% T(x+y) = T(x) + T(y) %}
- {% T(cx) = cT(x) %}
Matrix Representation
A transformation on a finite dimensional vector space is completely determined by its effect on each basis vector,
as demonstrated below.
{% T(\sum a_i e_i) = \sum a_i T(e_i) %}
Write the result of applying the transformation on the first basis vector as follows.
{%
T(\begin{bmatrix}
1 \\
0 \\
0 \\
\end{bmatrix}) = \begin{bmatrix}
a_1 \\
a_2 \\
a_3 \\
\end{bmatrix}
%}
The results of each application of the transformation on each basis vector can then be arranged in a matrix format.
{%
\begin{bmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2\\
a_3 & b_3 & c_3 \\
\end{bmatrix}
%}
This matrix, when applied to an arbitrary vector in the domain, will return the result of the linear transformation, represented
as a column vector.
Eigenvalues and vectors
Eigen Vectors/Eigen Values:
An eigenvector of a transformation is a vector that when transformed by the transformation, returns the same vector, possibly multiplited by a scaler.
The vector is referred to as an eigenvector, and the corresponding scale is called the eigenvalue.
{% A\vec{v} = a\vec{v} %}
Defintions
