Expression of A Basis
There are several ways to describe a basis for the space. Some are made ad hoc for a specific dimension. For example, there are several ways to give a basis in dim 3, like Euler angles.
The general case is to give a matrix with the components of the new basis vectors in columns. This is also the more general method because it can express any possible set of vectors even if it is not a basis. This matrix can be seen as three things:
Basis Matrix: Is a matrix that represents the basis, because its columns are the components of vectors of the basis. This matrix represents any vector of the new basis as linear combination of the current basis.
Rotation operator: When orthonormal bases are used, any other orthonormal basis can be defined by a rotation matrix. This matrix represents the rotation operator that rotates the vectors of the basis to the new one. It is exactly the same matrix as before because the rotation matrix multiplied by the identity matrix I has to be the new basis matrix.
Change of basis matrix: This matrix can be used to change different objects of the space to the new basis. Therefore is called "change of basis" matrix. It is important to note that some objects change their components with this matrix and some others, like vectors, with its inverse.
Read more about this topic: Basis (linear Algebra)
Famous quotes containing the words expression of a, expression of, expression and/or basis:
“A man finds room in the few square inches of the face for the traits of all his ancestors; for the expression of all his history, and his wants.”
—Ralph Waldo Emerson (1803–1882)
“Disease is a vital expression of the human organism.”
—Georg Groddeck (1866–1934)
“Wags try to invent new stories to tell about the legislature, and end by telling the old one about the senator who explained his unaccustomed possession of a large roll of bills by saying that someone pushed it over the transom while he slept. The expression “It came over the transom,” to explain any unusual good fortune, is part of local folklore.”
—For the State of Montana, U.S. public relief program (1935-1943)
“Self-alienation is the source of all degradation as well as, on the contrary, the basis of all true elevation. The first step will be a look inward, an isolating contemplation of our self. Whoever remains standing here proceeds only halfway. The second step must be an active look outward, an autonomous, determined observation of the outer world.”
—Novalis [Friedrich Von Hardenberg] (1772–1801)