Editing Change of basis (section)

==Change of basis formula==
Let <math>B_\mathrm {old}=(v_1, \ldots, v_n)</math> be a basis of a [[finite-dimensional vector space]] {{mvar|V}} over a [[field (mathematics)|field]] {{mvar|F}}.{{efn|Although a basis is generally defined as a set of vectors (for example, as a spanning set that is linearly independent), the [[tuple]] notation is convenient here, since the indexing by the first positive integers makes the basis an [[ordered basis]].}}

For {{math|1=''j'' = 1, ..., ''n''}}, one can define a vector {{math|''w''{{sub|''j''}}}} by its coordinates <math>a_{i,j}</math> over <math>B_\mathrm {old}\colon</math>
:<math>w_j=\sum_{i=1}^n a_{i,j}v_i.</math>

Let
:<math>A=\left(a_{i,j}\right)_{i,j}</math>
be the [[matrix (mathematics)|matrix]] whose {{mvar|j}}th column is formed by the coordinates of {{math|''w''{{sub|''j''}}}}. (Here and in what follows, the index {{mvar|i}} refers always to the rows of {{mvar|A}} and the <math>v_i,</math> while the index {{mvar|j}} refers always to the columns of {{mvar|A}} and the <math>w_j;</math> such a convention is useful for avoiding errors in explicit computations.)

Setting <math>B_\mathrm {new}=(w_1, \ldots, w_n),</math> one has that <math>B_\mathrm {new}</math> is a basis of {{mvar|V}} if and only if the matrix {{mvar|A}} is [[invertible matrix|invertible]], or equivalently if it has a nonzero [[determinant]]. In this case, {{mvar|A}} is said to be the ''change-of-basis matrix'' from the basis <math>B_\mathrm {old}</math> to the basis <math>B_\mathrm {new}.</math>

Given a vector <math>z\in V,</math> let <math>(x_1, \ldots, x_n) </math> be the coordinates of <math>z</math> over <math>B_\mathrm {old},</math> and <math>(y_1, \ldots, y_n) </math> its coordinates over <math>B_\mathrm {new};</math> that is 
:<math>z=\sum_{i=1}^nx_iv_i = \sum_{j=1}^ny_jw_j.</math>
(One could take the same summation index for the two sums, but choosing systematically the indexes {{mvar|i}} for the old basis and {{mvar|j}} for the new one makes clearer the formulas that follows, and helps avoiding errors in proofs and explicit computations.)

The ''change-of-basis formula'' expresses the coordinates over the old basis in terms of the coordinates over the new basis. With above notation, it is
:<math>x_i = \sum_{j=1}^n a_{i,j}y_j\qquad\text{for } i=1, \ldots, n.</math>

In terms of matrices, the change of basis formula is
:<math>\mathbf x = A\,\mathbf y,</math>
where <math>\mathbf x</math> and <math>\mathbf y</math> are the column vectors of the coordinates of {{mvar|z}} over <math>B_\mathrm {old}</math> and <math>B_\mathrm {new},</math> respectively.

''Proof:'' Using the above definition of the change-of basis matrix, one has
:<math>\begin{align}
z&=\sum_{j=1}^n y_jw_j\\
 &=\sum_{j=1}^n \left(y_j\sum_{i=1}^n a_{i,j}v_i\right)\\
 &=\sum_{i=1}^n \left(\sum_{j=1}^n a_{i,j} y_j \right) v_i.
\end{align}</math>

As <math>z=\textstyle \sum_{i=1}^n x_iv_i,</math> the change-of-basis formula results from the uniqueness of the decomposition of a vector over a basis.