Linear and Orthogonal Transformations

In mathematics, a transformation is a function that changes the coordinates of each point in a given set. There are two types of transformations: linear and orthogonal. In this article, we will discuss the differences between these two types of transformations, and how to apply them to your work!

What Are Transformations?

A transformation is a function that takes one set of coordinates to another. For example, we can transform the point $x=0$ to $y=-36$. This means we are changing the coordinate system.

Types Of Transformations

There are three types of transformations: linear, orthogonal, and nonlinear.

Linear transformations preserve the shape of objects, while orthogonal transformations keep the object’s size constant. Non-linear transformations distort the shape of objects.

Linear Transformation

A linear transformation L: V -> W is a rule that associates each vector x with another vector y. It satisfies two properties:

• Additivity/Homogeneity: If u and v are vectors from V and c is a scalar, then L(u + v) = Lu + Lv and L(cu) = cLu.
• Commutativity: L(u + v) = Lu + Lv and L(cu) = cLu.

Linear Transformation Examples

• Translation: L(x, y) = (x + a, y + b). This shifts each point (x,y) by vector v = (a ,b), resulting in the point at the same distance from the origin but offset.
• Rotation: The rotation power function is given by Rt(z) = z^t for all complex numbers z.
• Scaling: L(x,y) = (ax, by). This is a multiplication of each vector v by the scalar c from V to W.

Linear Transformation Matrix: 

A linear transformation matrix is a matrix that transforms a vector according to a linear equation. The most common type of linear transformation is scaling, which multiplies each component of the vector by a constant value. Other types of linear transformations include shearing and rotation.

Shearing

A shearing transformation distorts or skews a vector in some way. It is represented by a linear transformation matrix:

Shearing Matrix for a Vector Skewed Along the x-Axis:

(x) (y)

(−y) =.

Shearing Matrix for a Vector Skewed Along the y-Axis:

(x) (y)

(−x) = .

Rotation

A rotation is a linear transformation that rotates a vector around a fixed point. The fixed point is usually the origin, O, but it can be any other point in space. The angle of rotation, θ (in radians), determines how much the vector is turned. The direction of rotation, either clockwise or counterclockwise, determines which direction the vector is turned.

A rotation can be represented by a linear transformation matrix:

Rotation Matrix for a Clockwise Rotation about O:

cosθ −sinθ (x) (y)

sinθ cos θ = ; ϕ = angle of rotation in radians.

Rotation Matrix for a Counterclockwise Rotation about O:

cosθ −sinθ (x) (y)

sinθ cos θ =.

The above two matrices are the only possible rotations about the origin.

Orthogonal Transformation

A linear transformation T: Rn -> Rn that preserves the dot product between vectors is known as an orthogonal transformation. Such transformations are important in physics and engineering, where they are used to change coordinate systems.

There are several different types of orthogonal transformations. In this article, we will focus on the three most common types: rotation, reflection, and translation. Let’s start with rotation.

Rotation

A rotation is a transformation that rotates a vector around a fixed point in space. The most common type of rotation in a clockwise or counter-clockwise turn about the origin, O. However, any point P in space can be used as the centre of rotation.

The angle of rotation θ (in radians) determines how much the vector is turned. The direction of rotation, either clockwise or counter-clockwise, determines which side of the origin the vector ends upon. For example, a counter-clockwise turn about O will rotate a vector from quadrant I to quadrant II.

A rotation can be represented by an orthogonal matrix:

Rotation Matrix for a Clockwise Rotation about O:

(cosθ −sinθ) (x) (y)

(sinθ cos θ ) =.

There are also rotation matrices for counterclockwise rotations and rotations around P.

Rotation Matrix

Where ϕ is the angle of rotation in radians. Note that this matrix only applies to rotations about the origin O. If P is used as the centre of rotation, then a different matrix must be used.

Reflection

A reflection is a transformation that reflects a vector across a line or plane. The most common type of reflection is the mirror image, which reflects a vector over the x-axis (the real axis) or the y-axis (the imaginary axis). Other types of reflections include reflections across an arbitrary line and two intersecting lines.

The direction of the reflection is determined by the line or plane that is used as the mirror. For example, a vector reflected over the x-axis will be reflected from left to right, while a vector reflected over the y-axis will be reflected from top to bottom.

A reflection can be represented by an orthogonal matrix:

Reflection Matrix for a Vector Reflected Over the x-Axis:

(x) (y)

(−y) =.

Reflection Matrix for a Vector Reflected Over the y-Axis:

(x) (y)

(−x) = .

Translation

A translation is a transformation that moves a vector to a new location in space. The new location is specified by the vector Tx, Ty. Thus, the translation matrix can be written as:

Translation Matrix:

Tx Ty

=.

The translation matrix preserves the length of vectors and the direction of vectors pointing away from the origin. It also preserves angles between vectors and their scalar products (dot products).

Example: Rotation about O by 90° clockwise

Let’s consider a vector A that is rotated 90° clockwise about O, where θ = π/32 radians or 45°. The angle of rotation, θ, is given in radians.

A = (cosθ −sinθ)i + (sinθ cos θ )j

The rotated vector A’ can be found using the rotation matrix, R:

A’ = R(A)

= (cosθ −sinθ)i + (sinθ cos θ )j.

The rotated vector A’ is now pointing in the direction of j. Since it has been rotated 90° clockwise, it is pointing down and to the right instead of up and to the left as A was original.

Conclusion

Transformations are a powerful way to change your coordinates. By using linear or orthogonal transformations, you can easily change the direction of your object or reshape it in other ways. These transformations can help correct errors or alter your object to better suit your needs. Be sure to experiment with different transformation types to find the ones that work best for you.