We work out how the choice of basis affects the passage from linear maps to matrices. See DF page 419. Axler works this out, partly in the exercises, in Section 3D of LADR.
Duality
The dual vector space consists of linear maps from to the one-dimensional vector space . We write this vector space as ; it is called the dual space.
If is a linear map, there is an associated linear map
defined by .
Duality is an example of a contravariant functor.
If and then and . Then
If is the inclusion map of a subspace then is the restriction map from functions on to functions on .
If is the projection to the quotient space, then is the “extension” map that views a function on as a function on that vanishes on .
If is an “exact sequence” then the dual is also “exact”. The first map takes a function on and views it as a functional on vanishing on ; the second map restricts a function on to . So the kernel of the second map is the image of the first.
Proposition: The restriction map from to is surjective and the “extension” map from to is injective.
If is a linear map, then is an injective map where is the kernel of and is its range.