Linear transformations describe functions that take a vector and produce a new resulting vector; we may call it a function for vectors. Three views on vector exist:
- physics see it as a length (force) in a specific direction. This implies that the startpoint can vary and the vector remains equal.
- mathematicians add a root/ origin, equal for all vectors in the same space, and a “basis vector”, defining the coordinate system, to the view.
This implies that vectors have an identity, can be multiplied as given or with a scalar and added. Simplified addition and multiplication happens on/ with the “basis vector” (i + j).
- computer scientists describe the mathematical view via lists. Each list element describes the position in the given space.
Multiplying vectors can happen in two ways:
- scalar product, a · b = a * b * cos 𝛂 , with 𝛂 as angle between a and b
- vector product, a x b = a * b * sin 𝛂 , with 𝛂 as angle between a and b
Source https://en.wikipedia.org/wiki/Sine_and_cosine, adapted by author
- width = sin 𝛂 = b/c
- height = cos 𝛂 = a/c
Sin and cos are defined through the unit circle:
e = (cos a, sin a)
These operations are defined in the unit circle (radius = 1). For common angles compare https://en.wikipedia.org/wiki/Exact_trigonometric_values#Common_angles and https://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svg
Source: https://en.wikipedia.org/wiki/Transformation_matrix#/media/File:2D_affine_transformation_matrix.svg
Sub-clause: Eigenvectors are vectors that do not change the direction, in comparison to the given vector, after transformation. The eigenvalue is the scaling of the eigenvector.
If we use these transformation to represent motion, the following definitions are useful:
| Linear motion | Rotational motion | |
| displacement | s | angle, Φ, = s/ r |
| velocity | v = Δs / Δt | ω = ΔΦ / Δt = v / r |
| acceleration | a = Δv / Δt | 𝛼 = Δω / Δt = a / r |
| force | F | torque, = r * F |