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:
“Of course, some men are very effective caregivers [of elderly parents]. But this situation occurs far less frequently for males than females, because it is a role reversal. For women, caregiving is an expected duty; for men, it is an unexpected expression of love or devotion.”
—Tish Sommers (20th century)
“The psychological umbilical cord is more difficult to cut than the real one. We experience our children as extensions of ourselves, and we feel as though their behavior is an expression of something within us...instead of an expression of something in them. We see in our children our own reflection, and when we don’t like what we see, we feel angry at the reflection.”
—Elaine Heffner (20th century)
“The basis of shame is not some personal mistake of ours, but the ignominy, the humiliation we feel that we must be what we are without any choice in the matter, and that this humiliation is seen by everyone.”
—Milan Kundera (b. 1929)