Editing Linear algebra (section)

==Matrices==
{{Main|Matrix (mathematics)}}

Matrices allow explicit manipulation of finite-dimensional vector spaces and [[linear map]]s. Their theory is thus an essential part of linear algebra.

Let {{mvar|V}} be a finite-dimensional vector space over a field {{math|''F''}}, and {{math|('''v'''<sub>1</sub>, '''v'''<sub>2</sub>, ..., '''v'''<sub>''m''</sub>)}} be a basis of {{math|''V''}} (thus {{mvar|m}} is the dimension of {{math|''V''}}). By definition of a basis, the map
:<math>\begin{align}
(a_1, \ldots, a_m)&\mapsto a_1 \mathbf v_1+\cdots a_m \mathbf v_m\\
F^m &\to V
\end{align}</math>
is a [[bijection]] from {{math|''F<sup>m</sup>''}}, the set of the [[sequence (mathematics)|sequences]] of {{mvar|m}} elements of {{mvar|F}}, onto {{mvar|V}}. This is an [[isomorphism]] of vector spaces, if {{math|''F<sup>m</sup>''}} is equipped with its standard structure of vector space, where vector addition and scalar multiplication are done component by component.

This isomorphism allows representing a vector by its [[inverse image]] under this isomorphism, that is by the [[coordinate vector]] {{math|(''a''<sub>1</sub>, ..., ''a<sub>m</sub>'')}} or by the [[column matrix]]
:<math>\begin{bmatrix}a_1\\\vdots\\a_m\end{bmatrix}.</math>

If {{mvar|W}} is another finite dimensional vector space (possibly the same), with a basis {{math|('''w'''<sub>1</sub>, ..., '''w'''<sub>''n''</sub>)}}, a linear map {{mvar|f}} from {{mvar|W}} to {{mvar|V}} is well defined by its values on the basis elements, that is {{math|(''f''('''w'''<sub>1</sub>), ..., ''f''('''w'''<sub>''n''</sub>))}}. Thus, {{mvar|f}} is well represented by the list of the corresponding column matrices. That is, if 
:<math>f(w_j)=a_{1,j}v_1 + \cdots+a_{m,j}v_m,</math> 
for {{math|1=''j'' = 1, ..., ''n''}}, then {{mvar|f}} is represented by the matrix
:<math>\begin{bmatrix}
a_{1,1}&\cdots&a_{1,n}\\
\vdots&\ddots&\vdots\\
a_{m,1}&\cdots&a_{m,n}
\end{bmatrix},</math>
with {{mvar|m}} rows and {{mvar|n}} columns.

[[Matrix multiplication]] is defined in such a way that the product of two matrices is the matrix of the [[function composition|composition]] of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing the same concepts.

Two matrices that encode the same linear transformation in different bases are called [[similar (linear algebra)|similar]]. It can be proved that two matrices are similar if and only if one can transform one into the other by [[Elementary matrix|elementary row and column operations]]. For a matrix representing a linear map from {{mvar|W}} to {{mvar|V}}, the row operations correspond to change of bases in {{mvar|V}} and the column operations correspond to change of bases in {{mvar|W}}. Every matrix is similar to an [[identity matrix]] possibly bordered by zero rows and zero columns. In terms of vector spaces, this means that, for any linear map from {{mvar|W}} to {{mvar|V}}, there are bases such that a part of the basis of {{mvar|W}} is mapped bijectively on a part of the basis of {{mvar|V}}, and that the remaining basis elements of {{mvar|W}}, if any, are mapped to zero. [[Gaussian elimination]] is the basic algorithm for finding these elementary operations, and proving these results.