matrix representation of a bilinear form
Given a bilinear form, , we show how we can represent it with a matrix, with respect to a particular pair of bases for and
Suppose and are finite-dimensional and we have chosen bases, and . Now we define the matrix with entries . This will be the matrix associated to with respect to this basis as follows; If we write as column vectors in terms of the chosen bases, then check . Further if we choose the corresponding dual bases for and then and are the corresponding matrices for and , respectively (in the sense of linear maps). Thus we see that a symmetric bilinear form is represented by a symmetric matrix, and similarly for skew-symmetric forms.
Let and be new bases, and and the corresponding change of basis matrices. Then the new matrix is .
|Title||matrix representation of a bilinear form|
|Date of creation||2013-03-22 14:56:22|
|Last modified on||2013-03-22 14:56:22|
|Last modified by||vitriol (148)|