Page 115 - A Course in Linear Algebra with Applications

4.2: Vector Spaces and Subspaces 99

solution space of the homogeneous linear system AX = 0; it is
also known as the null space of the matrix A. (Question: why
is it necessary to have a homogeneous linear system here?)

Example 4.2.3
Let S denote the set of all real solutions y = y(x) of the
homogeneous linear differential equation

y" + by' + 6y = 0

defined in some interval [a, b]. Thus S is a subset of the vector
space C[a, &] of continuous functions on [a, b]. It is easy to
verify that S contains the zero function and that S is closed
with respect to addition and scalar multiplication; in other
words S is a subspace of C[a, b}.
The subspace S in this example is called the solution space
of the differential equation. More generally, one can define the
solution space of an arbitrary homogeneous linear differential
equation, or even of a system of such differential equations.
Systems of homogeneous linear differential equations are stud-
ied in Chapter Eight.

Linear combinations of vectors
Let vi, V2,..., v/; be vectors in a vector space V. If c\,
C2, .. , Cfc are any scalars, the vector
.
civi + c 2 v 2 H V c fcv fc
is called a linear combination of i, v 2 ,..., v^.
v
For example, consider two vectors in R 2

The most general linear combination of X\ and X 2 is