If is a row of vectors, each from a vector space , and if is a column of scalars of length , the product of the row of vectors with the column of scalars may be regarded as the pseudo matrix multiplication that evaluates to , which is an element of .
The vectors in span if every vector in may be written as for a least one column . The vectors in are linearly independent if a vector in may be written in the form in at most one way. Finally, the vectors in form a basis of if and only if every vector in may be written in the form in exactly one way. One calls the column of coordinates of the vector relative to the basis .
If is a basis of and is another basis of , then one has for exactly one column , and the matrix is invertible. One may write the pseudo matrix relations In this situation either or may be referred to as a change of coordinates or change of basis. Neither should be viewed as the matrix of a linear map. In this context the expressions and represent the same vector in if and only if the columns and are related by the formula .
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If is a linear map from a vector space to a vector space , is a basis of , and is a basis of , then one says that is the matrix of relative to and if one has whenever is the column of coordinates of a vector in relative to the basis and is the column of coordinates of relative to the basis .
There is a straightforward way of translating the previous description of the matrix of a linear map relative to given bases into the language of bases rather than, as above, the language of coordinates. For this one uses the notation to denote the row of vectors in . The row need not be a basis of . Since, however, is a basis of , one may write, for each subscript , for a unique column of scalars. Then the matrix of is the matrix , and, indeed, one has the equivalent pseudo matrix formula .
What happens to the matrix of a linear map when the bases are changed? Suppose , , , and . Since is linear, one sees that the first of these formulas leads to the relation . It then follows by elementary matrix arithmetic that .
When is a linear map from a vector space to itself, one usually uses a single basis in the roles of both and above.
When is a linear map from a vector space to itself, its matrix relative to a basis , and its matrix relative to a basis consisting of characteristic vectors of (if such a basis exists), then is a matrix whose diagonal entries are the proper values of and one has where .
Matrices can be used in at least three ways in this course:
The matrix of a linear map relative to given bases of the domain and the target spaces furnishes a definition of the linear map.
The matrix of change of basis for two given bases defines the relationship between the two bases.
A symmetric matrix defines a quadratic form.