5.2 Two–by–Two Matrices:
Linear Mappings
The
determinant
of a two–by–two matrix
is defined as
5.2.1 Specific Linear Mappings 1:
the Unit Matrix
This is the
trivial mapping represented by the
unit or
identity matrix,
,
We have
. Check that
for any vector
.
5.2.2 Specific Linear Mappings 2:
Stretching and Shrinking
These are linear mappings
represented by the multiples of the
unit matrix, where
is a real number such that
We have
. Check that in this case
for any vector
.
5.2.3 Specific Linear Mappings 3:
Projections
These are linear mappings
such as
We have
. Check that in this case, for any vector
,
: the vector is projected onto the
-axis.
5.2.4 Specific Linear Mappings 4:
Rotations
These are mappings
that rotate vectors around the origin by an angle
,
In this case,
. A vector
is rotated into
Examples for rotations are
Special Rotations:
In this case,
Special Rotations:
In this case,
5.2.5 Specific Linear Mappings 5:
Reflections
These are mappings
that reflect a vectors at a fixed axis:
In this case,
. A vector
is transformed into
Examples:
where we have a formula for trigonometric
functions (CHECK). Furthermore, we have
Sketch this in the
-
-plane (lecture). We recognise that
defines a reflection at the axis
defined by the direction of the vector
Special Reflection:
In this case,
Special Reflection:
In this case,