Page 159 - A Course in Linear Algebra with Applications

5.3: Operations with Subspaces 143

On writing down the polynomials with these coordinate vec-
3
2
3
tors, we obtain the basis -3x , x + 2x , x -5x 3 for U* + W*.
l
In the case of U D W the procedure is to find a basis for
U* Pi W*. This turns out to consist of the single vector

( ; )

Finally, read off that the polynomial

3
2
1 • (1 + 2x + x ) + 1 • (1 - x - x ) = 2 + x - x 2 + x 3
forms a basis of U l~l W.

Quotient Spaces
We conclude the section by describing another subspace
operation, the formation of the quotient space of a vector
space with respect to a subspace. This new vector space is
formed by identifying the vectors in certain subsets of the
given vector space, which is a construction found throughout
algebra.
Proceeding now to the details, let us consider a vector
space V with a fixed subspace U. The first step is to define
certain subsets called cosets: the coset of U containing a given
vector v is the subset of V

u
v + [/ = {v + |ue[/}.

Notice that the coset v + U really does contain the vector v
since v = v + OEv + U. Observe also that the coset v + U
can be represented by any one of its elements in the sense that
(v + u) + U = v + U for all u G U.
An important feature of the cosets of a given subspace is
that they are disjoint, i.e., they do not overlap.