This set is due Friday, March 12.
It is common in algebraic settings to build new structures by taking quotients of old ones. This occurs in topology (building quotient spaces), in abstract algebra when building field extensions, or in the homomorphism theorems. Here we explore quotients in vector spaces.
First we briefly consider an example from differential equations.
Let be the space consisting of all continuously differentiable functions and let be the space of all continuous functions Let be the linear transformation
Show that is a real vector space of dimension 1, by showing that iff is a constant. This means, of course, that
Show that by finding a particular solution to the equation One way of doing this is by looking for such a function of the form for some constant Find the form of an arbitrary function such that by noting that if then
More generally, show that is surjective, by finding for any the explicit form of the solutions to the equation It may help you solve this equation if you first multiply both sides by
For another example, denote by the space of all matrices with real entries. Define a map by Show explicitly that has dimension 6 and that is surjective.
Now we abstract certain features of these examples to a general setting:
Suppose is a field and is a linear transformation between two -vector spaces and It is not necessary to assume that or are finite dimensional.
Let and let be any preimage, i.e., Show that the set of all preimages of is precisely
Define a relation in by setting iff Show that is an equivalence relation. Denote by the equivalence class of the vector
Let be the quotient of by i.e., the collection of equivalence classes of the relation We want to give the structure of an -vector space. In order to do this, we define and for all and Show that this is well-defined and satisfies the axioms of an -vector space. What is the usual name we give to the 0 vector of this space?
It is standard to denote by Define two functions and as follows: is given by Also, is given by Show that is well-defined and that both and are linear.
Show that that is a surjection, and that is an isomorphism between and In particular, any surjective image of a vector space by a linear map can be identified with a quotient of