Connection between Matrices and Geometric Transformations
A matrix defines a linear transformation of the Cartesian plane. It always maps the origin to itself, so a pure translation needs vector addition or homogeneous coordinates instead.
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A matrix defines a linear transformation of the Cartesian plane. It always maps the origin to itself, so a pure translation needs vector addition or homogeneous coordinates instead.
A point in the Cartesian plane, often symbolized by the ordered pair , can also be symbolized by the position vector . This position vector notation will be frequently used in discussing the connection between matrices and transformations.
If a point is transformed by the matrix , its image is obtained from matrix multiplication:
and .
If represents any point in the Cartesian plane, find the product of .
Solution:
The matrix product is:
The matrix maps to . This coordinate rule describes a counter-clockwise rotation about the origin.
Find the image of , with vertices , , and transformed by the matrix .
Solution:
First, we can write the coordinates of the points as columns of a matrix, i.e., . The three columns correspond to , , and , in that order.
Next, multiply this matrix from the left by .
The result of the transformation is a new triangle with vertices , , and .
The matrix represents a rotation about the origin.
The point is transformed into .
This is a clockwise rotation, equivalently a counter-clockwise rotation, about the origin.
Transformation matrix .
Vertices: , , .
Published: . Updated: .
Point matrix: .
Image vertices: , , .
This is a horizontal shear: the -coordinate changes by twice the -coordinate, while remains unchanged.