Page 163 - A Course in Linear Algebra with Applications
P. 163
5.3: Operations with Subspaces 147
Our last example of coset formation is a geometric one.
Example 5.3.10
Let A and B be vectors in R 3 representing non-parallel line
segments in 3-dimensional space. Then the subspace
U=<A, B>
has dimension 2 and consists of all cA + dB, (c, d E R). The
vectors in U are represented by line segments, drawn from the
origin, which lie in a plane P. Now choose X G R , with
T
X = (xi, x 2, x 3) .
A typical vector in the coset X + U has the form
X + cA + dB, with c, d e R, i.e.,
T
(xi + cai + dbi x 2 + ca 2 + db2 X3 + ca 3 + db 3) .
(
Now the points xi+cai+d&i, X2+ca2+db 2, Xs + cas + db 3)
lie in the plane Pi passing through the point (xi, x 2, £3),
which is parallel to the plane P. This is seen by forming the
line segment joining two such points. The elements of X + U
correspond to the points in the plane Pi: the latter is called
a translate of the plane P.
Dimension of a Quotient Space
We conclude the discussion by noting a simple formula
for the dimension of a quotient space of a finite dimensional
vector space.
Theorem 5.3.7
Let U be a subspace of a finite dimensional vector space V.
Then
dim(V/C/) = dim(y) - dim(£/).