Glossary
The following are terms to know in alphabetical order (not in an order that you should learn these concepts)
Basis Vectors
(Linearly Independent) Vectors that can be used to span the space of a coordinate system.
Example: The î vector in the X direction and the ĵ vector in the Y direction, are the basis vectors of of the XY coordinate system. You can specify coordinates in terms of the sum of 2 scalars each multiplying with a basis vector (Linear Combination). By having different Basis vectors, you can create alternate coordinate spaces
Coordinate System
Linear Combination
When you scale 2 vectors separately then add them together to get a vector output.
Linear Composition / Product
A combination of 2 Linear transformations in a specific sequence, which can be encoded itself as a singular linear transformation.
- Example: Combining a rotation, then a shear transformation, as a single transformation that applies both effects
Numerically, a composition of 2 transformations can be found by multiplying the 2 matrices together (where the first transformation is on the right side of the multiplication)
Linear Dependence / Independence
A vector is said to be linearly dependent if it could be added as a "basis" vector to an existing set of basis vector of a coordinate space, and not increase dimensionality. In other words, a dependent vector is already within the span of the basis vectors. Linear independence is the opposite, in that the vector was not part of the span and thus adding it as a basis vector allows for the span to increase in dimensionality
Linear Transformation
A function that transforms vector into a single other vector. Since they are used to map any number of input vectors to output vectors, we can use them to transform the space itself,
A transformation is only linear if the resultant space:
- Does not have its origin moved
- If all previously equidistant points remain equidistant
- parallel lines before remain parallel after
- Straight lines must remain straight
A Linear transformation can be described as a set of vectors that describe where the initial i and j (and k) basis vectors will land as a result of the transformation, they are often kept in an NxN matrix.
Scalar
A non-vector, as in does not specify a direction. At most can only define a magnitude.
Scalar Multiplication with a Vector
Takes a scalar and a vector, and outputs a vector with the same number of dimensions. Commutative,
Geometric: Scales the magnitude of a vector without changing the direction (except in the case of negative scalars where the direction reverses)
Numeric: Multiply each element of a vector with the scalar to get an output,
Span
The set of all achievable vectors retrievable from every possible Linear Combination of a set of defined vectors. If the input vectors are on the same line / within the span of the other vectors, the dimensionality of the result will be reduced!
Vector
Geometric: A direction in space, where the numbers are the translation to go from the tail end of the vector to the head.
Numeric: An ordered set of numbers.
Vector Addition
Takes 2 vectors and outputs a vector.. Commutative. Both input vectors must have the same number of dimensions, Output vector will have the same number of dimensions.
Geometric: The result of following the motion of both input vectors, one after another
Numeric: Add like terms of the input to get the output.