$N=BMB_{−1}$

- $M$: matrix in the old basis
- $N$: matrix in the new basis
- $B$: change of basis matrix

The change of basis matrix $C$ is the matrix that allows us to express the new basis in an old basis:

$x_{old}=Cx_{new}$

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:

$x_{new}y_{new} =1x_{old}+2y_{old}=3x_{old}+4y_{old} $

$[x_{new}y_{new} ]=[13 24 ][x_{old}y_{old} ]$

$M=[13 24 ]$

$x_{new} =Mx_{old} $

The usual question then is: given a vector in the new basis, how do we represent it in the old basis?

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:

$N=BMB_{−1}=OMO_{T}$

Every vector space is defined over a field.

E.g. in $R_{3}$, the underlying field is $R$, the real numbers. And in $C_{2}$ the underlying field is $C$, 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 $R_{3}$, the underlying field is $R$, and a scalar is a member of $R$, 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.