Page 114 - A Course in Linear Algebra with Applications

98 Chapter Pour: Introduction to Vector Spaces

where t is an arbitrary real number. Since

2 2t
M + ( A = ( 2(*i+< )
2

and

—3t J \ —3ct

S is closed under addition and scalar multiplication; also S
contains the zero vector I 1, as may be seen by taking t to

2
be to 0. Hence S is a subspace of R .
In fact this subspace has geometrical significance. For
/ 2 A
an arbitrary vector I 1 of S may be represented by a line
segment in the plane with initial point the origin and end point
(2t,—3t). But the latter is a general point on the line with
equation 3x + 2y = 0. Therefore the subspace S corresponds
to the set of line segments drawn from the origin along the
line 3x + 2y = 0.

Example 4.2.2
This example is an important one. Consider the homogeneous
linear system
AX = 0

in n unknowns over some field of scalars F and let S denote
the set of all solutions of the linear system, that is, all the
n-column vectors X over F that satisfy AX — 0. Then S is a
n
subset of F and it certainly contains the zero vector. Now if
X and Y are solutions of the linear system and c is any scalar,
then

A(X + Y) = AX + AY = 0 and A(cX) = c(AX) = 0.

Thus X + Y and cX belong to S and it follows that S is a
n
subspace of the vector space R . This subspace is called the