Page 162 - A Course in Linear Algebra with Applications

146 Chapter Five: Basis and Dimension

At the opposite extreme, we could take U = V. Now
V/V consists of the cosets v + V = V, i.e., there is just one
element. So V/V is a zero vector space.

We move on to more interesting examples of coset forma-
tion.

Example 5.3.9

Let S be the set of all solutions of a consistent linear system
AX = B of m equations in n unknowns over a field F. If
n
5 = 0, then S is a subspace of F , namely, the solution
space U of the associated homogeneous linear system AX = 0.
However, if B ^ 0, then S is not a subspace: but we will see
that it is a coset of the subspace U.
Since the system is consistent, there is at least one solu-
tion, say Xi. Suppose X is another solution. Then we have
AX\ = B and AX = B. Subtracting the first of these equa-
tions from the second, we find that

0 = AX - AX l = A(X - Xi),

so that X — Xi E U and X £ Xi + U, where U is the solution
space of the system AX = 0. Hence every solution of AX = B
belongs to the coset X\ + U and thus S C X\ + U.
Conversely, consider any Y G X\ + U, say Y = X\ + Z
where Z eU. Then AY = AX l+AZ = B+0 = B. Therefore
Y ES SindS = X 1 + U.
These considerations have established the following re-
sult.
Theorem 5.3.6
Let AX — B he a consistent linear system. Let X\ he any
fixed solution of the system and let U he the solution space of
the associated homogeneous linear system AX = 0. Then the
set of all solutions of the linear system AX = B is the coset
X^ + U.