Last modified on 25 May 2010, at 18:36

Linear Algebra/Computing Linear Maps

Linear Algebra
 ← Rangespace and Nullspace Computing Linear Maps Representing Linear Maps with Matrices → 

The prior section shows that a linear map is determined by its action on a basis. In fact, the equation


h(\vec{v})
=h(c_1\cdot\vec{\beta}_1+\dots+c_n\cdot\vec{\beta}_n)
=c_1\cdot h(\vec{\beta}_1)+\dots +c_n\cdot h(\vec{\beta}_n)

shows that, if we know the value of the map on the vectors in a basis, then we can compute the value of the map on any vector \vec{v} at all. We just need to find the c's to express \vec{v} with respect to the basis.

This section gives the scheme that computes, from the representation of a vector in the domain {\rm Rep}_{B}(\vec{v}), the representation of that vector's image in the codomain {\rm
  Rep}_{D}(h(\vec{v})), using the representations of 
h(\vec{\beta}_1) , ...,  h(\vec{\beta}_n) .

Linear Algebra
 ← Rangespace and Nullspace Computing Linear Maps Representing Linear Maps with Matrices →