Linear Algebra from Elementary to Advanced Course 1: Linear Systems and Matrix Equations
3.3 Matrices and Linear Transformations
Objectives
★ Find the standard matrix of a linear transformation.
★ Identify whether a linear transformation is onto and one-to-one.
Definitions
●
The standard matrix for the linear transformation
for all in the domain of .
●
A transformation
least one vector in
●
is onto
in
if each vector
in
such that
is the image
of at
.
A transformation
one vector
is the matrix
is one-to-one if each vector
. In other words,
is onto when
in
is the image of at most
whenever
.
Theorems
The Standard Matrix of a Linear Transformation
Let
for all
to
in
be a linear transformation. Then there is a unique matrix
such that
.
This matrix is the
matrix whose th column is the vector
column of the
identity matrix:
, where
is the th
One-to-One Transformations
The linear transformation
the trivial solution.
to
is one-to-one if and only if the equation
has only
Linear Algebra from Elementary to Advanced Course 1: Linear Systems and Matrix Equations
One-to-One and Onto Transformations and Matrices
Let
be the standard matrix for the linear transformation
to
. Then
maps
onto
if and only if the columns of span
●
is one-to-one if and only if the columns of are linearly independent.
●
Common Linear Transformations and Matrices
The following tables list some linear transformations
and their standard matrices.
Reflections
Transformation
Reflection through the
Standard Matrix
-axis
Reflection through the -axis
Reflection through the line
Reflection through the line
Reflection through the origin
Contractions and Expansions
Transformation
Horizontal contraction or expansion by a factor
of
Vertical contraction or expansion by a factor of
Standard Matrix
Linear Algebra from Elementary to Advanced Course 1: Linear Systems and Matrix Equations
Shears
Transformation
Standard Matrix
Horizontal shear by a factor of
Vertical shear by a factor of
Projections
Transformation
Projection onto the
Standard Matrix
-axis
Projection onto the -axis
Other Resources
You can further explore this lesson’s topics at the following links.
● Webpage: Interactive Linear Algebra - 3.1 Matrix Transformations
● Webpage: Interactive Linear Algebra - 3.2 One-to-One and Onto Transformations
● Webpage: Rotation Matrices Explained (Data Genetics Blog)