Vector space

where:

The change of basis matrix $C$ 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 $(x_{old},y_{old})$. 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 $M$:

That $M^{-1}$ 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 ${\mathbb{R}}^3$, the underlying field is $\mathbb{R}$, the real numbers. And in ${\mathbb{C}}^2$ the underlying field is $\mathbb{C}$, 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 ${\mathbb{R}}^3$, the underlying field is $\mathbb{R}$, and a scalar is a member of $\mathbb{R}$, i.e. a real number.