Game Math: Vectors, Linear Independence, and Basis

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Parallel vectors cannot represent a coordinate system.

For example, if you're trying to represent 3D space and all the XYZ axes are parallel to each other, you cannot express a specific position.

In this case, the parallel XYZ axes are said to be in a state of linear dependence.

Conversely, when they are not parallel, they are linearly independent — and the XYZ 3D axes we typically work with are in a linearly independent state.

Each of these XYZ axes is called a basis vector (and when they are also unit vectors, they are called standard basis vectors).

The complete set is called a basis.

Building on this: the number of basis vectors that constitute a coordinate system's basis defines its dimensionality.