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:
“They were making their way with the resigned expression of those who are condemned to hope forever.”
—Charles Baudelaire (18211867)
“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 Lifes idea, Truth and Truths idea, never make men sick, sinful, or mortal.”
—Mary Baker Eddy (18211910)
“My dream is that as the years go by and the world knows more and more of America, it ... will turn to America for those moral inspirations that lie at the basis of all freedom ... that America will come into the full light of the day when all shall know that she puts human rights above all other rights, and that her flag is the flag not only of America but of humanity.”
—Woodrow Wilson (18561924)