where:
  • : matrix in the old basis
  • : matrix in the new basis
  • : change of basis matrix
The change of basis matrix is the matrix that allows us to express the new basis in an old basis:
Mnemonic is as follows: consider we have an initial basis . Now, we define the new basis in terms of the old basis, e.g.:
which can be written in matrix form as:
and so if we set:
we have:
The usual question then is: given a vector in the new basis, how do we represent it in the old basis?
The answer is that we simply have to calculate the matrix inverse of :
That is the matrix inverse.
When we have a symmetric matrix, a change of bases keeps symmetry iff it is done by an orthogonal matrix, in which case:
Every vector space is defined over a field.
E.g. in , the underlying field is , the real numbers. And in the underlying field is , the complex numbers.
Any field can be used, including finite field. But the underlying thing has to be a field, because the definitions of a vector need all field properties to hold to make sense.
Elements of the underlying field of a vector space are known as scalar.
A member of the underlying field of a vector space. E.g. in , the underlying field is , and a scalar is a member of , i.e. a real number.

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Vector space is a very important subject about which there is a lot to say.
For example, this sentence. And then another one.