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, expression and/or basis:
“A wicked mortal is not the idea of God. He is little else than the expression of error. To suppose that sin, lust, hatred, envy, hypocrisy, revenge, have life abiding in them, is a terrible mistake. Life and Life’s idea, Truth and Truth’s idea, never make men sick, sinful, or mortal.”
—Mary Baker Eddy (1821–1910)
“The American adolescent, then, is faced, as are the adolescents of all countries who have entered or are entering the machine age, with the question: freedom from what and at what price? The American feels so rich in his opportunities for free expression that he often no longer knows what it is he is free from. Neither does he know where he is not free; he does not recognize his native autocrats when he sees them.”
—Erik H. Erikson (1904–1994)
“Compassion has no place in the natural order of the world which operates on the basis of necessity. Compassion opposes this order and is therefore best thought of as being in some way supernatural.”
—John Berger (b. 1926)