Linear Algebra

Vector Space

A vector space, a.k.a linear space, is a set of objects called vectors, which may be added together and multiplied (scaled) by numbers called scalars. A quantity described by multiple scalars, such as having both magnitude and direction, is called a vector. Scalars in linear algebra are real numbers or generally elements of a field that relate to vectors in an associated vector space through the operation of scalar multiplication.

Generally, a vector space may be defined by using any field instead of real numbers, such as complex numbers.

Linear Subspace

A linear subspace, a.k.a vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.

Transform Matrix

Rotation Matrix

A rotation matrix is a type of transform matrix that is used to perform a rotation in Euclidean space . The following matrix will rotate the points in the xy plane counter-clockwise through an angle $\theta$ with respect to the x-axis about the origin in a two-dimensional Cartesian coordinate system .

$$ Rot(\theta) = \begin{bmatrix} \cos(\theta)\quad -\sin(\theta)\\ \sin(\theta)\qquad \cos(\theta) \end{bmatrix} $$