# Change of basis for a vector space

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Related KMR pages:

Shift of Basis (in general).
Representation and reconstruction of vectors.

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Other relevant sources of information:

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The Basis-Shift formula for Vectors

The Base-Shift formula for Numbers carries over verbatim to the Basis-Shift formula for Vectors, except for the replacement of the term $\, B_{ase} \,$ by the term $\, B_{asis} \,$ and the replacement of the term $\, d_{igits} \,$ by the term $\, c_{oordinates} \,$.

It is often useful to have a symbolic name for the operation of shifting the $\, B_{asis} \,$ of a $\, v_{ector} \,$ from $\, {B_{asis}}_1 \,$ to $\, {B_{asis}}_2 \,$. This operation will be called $\, B_{asisShift} \,$ and its action on a given $\, v_{ector} \,$ will be expressed in three different ways:

i) $\, {[B_{asisShift}]}_{{B_{asis}}_2}^{{B_{asis}}_1} \,$ or

ii) $\, {[v_{ector}]}_{{B_{asis}}_2}^{{B_{asis}}_1} \,$ if the $\, B_{asisShift} \,$ operation is clear from the context, or

iii) $\, {[\,\,\,]}_{{B_{asis}}_2}^{{B_{asis}}_1} \,$ if both the $\, v_{ector} \,$ and the $\, B_{asisShift} \,$ operation are clear from the context.

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$U \triangleq \begin{bmatrix} | & & | \\ u_1 & \cdots & u_m \\ | & & | \end{bmatrix}, \; U' \triangleq \begin{bmatrix} | & & | \\ u'_1 & \cdots & u'_m \\ | & & | \end{bmatrix} \,$,

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$x \equiv \left( [x]_U \right)_U \equiv x_1 \, u_1 + \cdots + x_m \, u_m \equiv \left( [x]_{U'} \right)_{U'} \equiv x'_1 \, u'_1 + \cdots + x'_m \, u'_m \,$

Interpreting these relationships in matrix algebra leads to the matrix identities:

$\, x \equiv U \begin{bmatrix} x_1 \\ \vdots \\ x_m \end{bmatrix} \equiv \begin{bmatrix} | & & | \\ u_1 & \cdots & u_m \\ | & & | \end{bmatrix} \begin{bmatrix} x_1 \\ \vdots \\ x_m \end{bmatrix} \equiv$

$\,\;\; \equiv U' \begin{bmatrix} x'_1 \\ \vdots \\ x'_m \end{bmatrix} \equiv \begin{bmatrix} | & & | \\ u'_1 & \cdots & u'_m \\ | & & | \end{bmatrix} \begin{bmatrix} x'_1 \\ \vdots \\ x'_m \end{bmatrix}$,

and we arrive at the matrix equality:

$[x]_U \equiv U^{-1}U'[x]_{U'} \,$.

Hence, multiplying the (column) coordinate vector $\, [x]_{U'} \,$ with the matrix $\, U^{-1}U' \,$ gives the (column) coordinate vector $\, [x]_U \,$, and the matrix that shifts the basis of a vector space from $\, U' \,$ to $\, U \,$ is given by:

${[B_{asisShift}]}_U^{U'} \equiv U^{-1}U' \,$.

In matrix algebra, we can therefore the express the operation of change of basis from $\, U' \,$ to $\, U \,$ as:

$[x]_U \equiv {[B_{asisShift}]}_U^{U'} [x]_{U'} \,$.

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Shift of basis for vectors:

Assume that $\, \mathbb{C}^m \,$ is an inner product space over the complex numbers $\, \mathbb{C} \,$
with the inner product given by $\, x \cdot y = x_1 \overline{y_1} + \cdots + x_m \overline{y_m} \,$ for $\, x, y \in \mathbb{C}^m$.

Then we have the respective representations

$\, [x]_B = { \left< \hat{x} \right> }_B = { \left< \begin{matrix} \hat{x}_1 \\ \vdots \\ \hat{x}_m \end{matrix} \right> }_B \,$ and $\, [x]_{B'} = { \left< \hat{x'} \right> }_{B'} ={ \left< \begin{matrix} \hat{x'}_1 \\ \vdots \\ \hat{x'}_m \end{matrix} \right> }_{B'}$,

and the respective reconstructions

$\, x = \left( [x]_B \right)_B = \hat{x}_1 \, b_1 + \cdots + \hat{x}_m \, b_m = \left( [x]_{B'} \right)_{B'} = \hat{x'}_1 \, b'_1 + \cdots + \hat{x'}_m \, b'_m$.

In matrix notation, we can write

$\, [x]_{B'} = [ \;\; ]_{B'}^B \, [x]_B$.

$\, [x]_B = [ \;\; ]_B^{B'} \, [x]_{B'} = [ \;\; ]_B^{B'} \, [ \;\; ]_{B'}^B \, [x]_B$.

Hence we have

$\, [ \;\; ]_B^{B'} \, [ \;\; ]_{B'}^B = I$,

where $\, I \,$ is the identity matrix, and therefore

$\, [ \;\; ]_B^{B'} = { [ \;\; ]_{B'}^B }^{-1}$.

Now, if $\, [ \;\; ]_{B'}^B \,$ is a unitary matrix, we have

$\, { [ \;\; ]_{B'}^B }^{-1} = { [ \;\; ]_{B'}^B }^{*}$,

where the * denotes “complex transposition” of the matrix elements,
which means transposition of them accompanied by their complex conjugation.

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Representation and Reconstruction of Vectors 1:

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Change of basis for vectors:

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Numbers and Vectors are naturally related to each other:

Hence Numbers and Vectors are naturally equivalent.

Numbers and Vectors are naturally related to Music:

Check out our section on Category Theory

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