Editing Basis (linear algebra) (section)

== Change of basis ==
{{main|Change of basis}}
Let {{math|''V''}} be a vector space of dimension {{mvar|n}} over a field {{math|''F''}}. Given two (ordered) bases <math>B_\text{old} = (\mathbf v_1, \ldots, \mathbf v_n)</math> and <math>B_\text{new} = (\mathbf w_1, \ldots, \mathbf w_n)</math> of {{math|''V''}}, it is often useful to express the coordinates of a vector {{mvar|x}} with respect to <math>B_\mathrm{old}</math> in terms of the coordinates with respect to <math>B_\mathrm{new}.</math> This can be done by the ''change-of-basis formula'', that is described below. The subscripts "old" and "new" have been chosen because it is customary to refer to <math>B_\mathrm{old}</math> and <math>B_\mathrm{new}</math> as the ''old basis'' and the ''new basis'', respectively. It is useful to describe the old coordinates in terms of the new ones, because, in general, one has [[expression (mathematics)|expressions]] involving the old coordinates, and if one wants to obtain equivalent expressions in terms of the new coordinates; this is obtained by replacing the old coordinates by their expressions in terms of the new coordinates.

Typically, the new basis vectors are given by their coordinates over the old basis, that is, 
<math display="block">\mathbf w_j = \sum_{i=1}^n a_{i,j} \mathbf v_i.</math>
If <math>(x_1, \ldots, x_n)</math> and <math>(y_1, \ldots, y_n)</math> are the coordinates of a vector {{math|'''x'''}} over the old and the new basis respectively, the change-of-basis formula is 
<math display="block">x_i = \sum_{j=1}^n a_{i,j}y_j,</math>
for {{math|1=''i'' = 1, ..., ''n''}}.

This formula may be concisely written in [[matrix (mathematics)|matrix]] notation. Let {{mvar|A}} be the matrix of the {{nowrap|<math>a_{i,j}</math>,}} and
<math display="block">X= \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} \quad \text{and} \quad Y = \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix}</math>
be the [[column vector]]s of the coordinates of {{math|'''v'''}} in the old and the new basis respectively, then the formula for changing coordinates is
<math display="block">X = A Y.</math>

The formula can be proven by considering the decomposition of the vector {{math|'''x'''}} on the two bases: one has 
<math display="block">\mathbf x = \sum_{i=1}^n x_i \mathbf v_i,</math>
and
<math display="block">\mathbf x =\sum_{j=1}^n y_j \mathbf w_j
= \sum_{j=1}^n y_j\sum_{i=1}^n a_{i,j}\mathbf v_i
= \sum_{i=1}^n \biggl(\sum_{j=1}^n a_{i,j}y_j\biggr)\mathbf v_i.</math>

The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here {{nowrap|<math>B_\text{old}</math>;}} that is
<math display="block">x_i = \sum_{j=1}^n a_{i,j} y_j,</math>
for {{math|1=''i'' = 1, ..., ''n''}}.