Change of Basis
Overview
A vector in a vector space can be written as a sum over a basis of the vector space.
{% \vec{x} = a_1 v_1 + ... a_n v_n %}
Here {% v_1, ..., v_n %} are basis vectors. The vector is often identified with a column vector as such:
{%
\begin{bmatrix}
a_1 \\
... \\
a_n \\
\end{bmatrix}
%}
If a new basis for the vector space is chosen, {% w_1, ... ,w_n %}, then the
coeficients of the new basis vectors which represent the vector will be different. In fact, the new column vector will be equal to
a matrix (representing the change of basis) times the original column vector.
{% \vec{v}' = M \vec{v} %}
Each column of the change of basis matrix represents one of the new basis vectors expressed as a sum over the old basis vectors.
For example, the following multiplication is equal to the first column of the change of basis matrix {% M %}
{% M \times \begin{bmatrix}
1 \\
0 \\
... \\
0 \\
\end{bmatrix}
%}
Inverse Change of Basis
Starting with a change of basis (we use the
Dirac notation
for vectors)
{% |i'> = \sum_j U_{ji} |j> %}
The there is an inverse change of basis which reverts back to the original basis set.
{% |i> = \sum_j V_{ji} |j'> %}
The elements of the inverse change can be calculated
{% V_{ki} = <k'|i> = (<i|k'>)^{*} = U_{ik}^* = (U ^\dagger)_{ki} %}
This implies that
{% |i> = U^{\dagger}U |i> \rightarrow U^{\dagger}U = 1 %}
Linear Transformations
When a linear transformation (matrix) is specified for a given basis, the equivalent matrix operating on column vectors stated in terms of a
new basis is easily calculated.
{% A' = M^{-1} A M %}
That is, first perform a change of basis to the new basis. Then operate with the linear transformation, then transform back.
