Matrix representations of geometric transformations in the plane, including images of basis vectors under a 2x2 matrix, enlargements and horizontal or vertical stretches, rotations, reflections, translations, composition of transformations using matrix products, and area scale factor ∣detM∣.
A (2×2) matrix M=(acbd) represents a transformation of points in the cartesian plane. For such transformations, we consider x-coordinates in vector form (10), and y-coordinates as (01):
In general, the point (x,y) is transformed to
We call the input to a transformation the object and the output the image.
When a transformation M is applied to a shape, the area of the image is
The absolute value is there as a negative determinant flips the orientation of the object, but that does not change the area.
The matrix M=(k00k) acts as a geometric enlargement with a scale factor k.
Try adjusting k or the position of the gray square in the simulation below.
(k001)
(100k)
The matrix M=(k001) acts as a geometric stretch with a scale factor k in the horizontal direction.
The matrix M=(100k) acts as a geometric stretch with a scale factor k in the vertical direction.
We can represent rotations about the origin with matrices:
A counterclockwise rotation is represented by (cosθsinθ−sinθcosθ).
A clockwise rotation is represented by (cosθ−sinθsinθcosθ).
![<ul>
<li>A 2D Cartesian coordinate system with horizontal and vertical axes, each shown with arrowheads.</li>
<li>Two congruent polygonal shapes sharing a common vertex at the origin: one gray (positioned in the first quadrant, with a short horizontal top segment and a slanted segment down to the origin) and one purple (rotated into the fourth/third quadrants).</li>
<li>An orange dashed arc with an arrow, drawn clockwise around the origin, labeled θ, indicating the angular relation between the two shapes.</li>
<li>An orange 2×2 matrix displayed at lower left: [ [cos θ, sin θ], [−sin θ, cos θ] ].</li>
</ul>](https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/cheat_sheet%2F1755981851489-c.png?alt=media&token=ef29db09-a780-4111-9009-58ed6c34ee28)
The matrix (cos2θsin2θsin2θ−cos2θ) represents a reflection in the line y=xtanθ, which is the line through the origin forming an angle of θ with the positive x-axis.
In summary:
If a point P is translated by a vector (ab), apply a translation a units to the right and b units up:
Geometric transformations represented by matrices can be chained together, and the combined transformation is represented by the product of the matrices. For example, transformation A then B then C is represented by the matrix (CBA).
Notice that the matrix product has the opposite order from the transformations, since
Nice work completing Geometric Transformations with Matrices, here's a quick recap of what we covered:
Exercises checked off
Matrix representations of geometric transformations in the plane, including images of basis vectors under a 2x2 matrix, enlargements and horizontal or vertical stretches, rotations, reflections, translations, composition of transformations using matrix products, and area scale factor ∣detM∣.
A (2×2) matrix M=(acbd) represents a transformation of points in the cartesian plane. For such transformations, we consider x-coordinates in vector form (10), and y-coordinates as (01):
In general, the point (x,y) is transformed to
We call the input to a transformation the object and the output the image.
When a transformation M is applied to a shape, the area of the image is
The absolute value is there as a negative determinant flips the orientation of the object, but that does not change the area.
The matrix M=(k00k) acts as a geometric enlargement with a scale factor k.
Try adjusting k or the position of the gray square in the simulation below.
(k001)
(100k)
The matrix M=(k001) acts as a geometric stretch with a scale factor k in the horizontal direction.
The matrix M=(100k) acts as a geometric stretch with a scale factor k in the vertical direction.
We can represent rotations about the origin with matrices:
A counterclockwise rotation is represented by (cosθsinθ−sinθcosθ).
A clockwise rotation is represented by (cosθ−sinθsinθcosθ).
The matrix (cos2θsin2θsin2θ−cos2θ) represents a reflection in the line y=xtanθ, which is the line through the origin forming an angle of θ with the positive x-axis.
In summary:
If a point P is translated by a vector (ab), apply a translation a units to the right and b units up:
Geometric transformations represented by matrices can be chained together, and the combined transformation is represented by the product of the matrices. For example, transformation A then B then C is represented by the matrix (CBA).
Notice that the matrix product has the opposite order from the transformations, since
Nice work completing Geometric Transformations with Matrices, here's a quick recap of what we covered:
Exercises checked off
On IB exams, you will not be expected to
IB exams will never ask you to
On IB exams, you will not be expected to
IB exams will never ask you to