Page 121 - A Course in Linear Algebra with Applications

4.3: Linear Independence in Vector Spaces 105

Linear dependence in R 3

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Consider three vectors A, B,C in Euclidean space R ,
and represent them by line segments in 3-dimensional space
with a common initial point. If these vectors form a linearly
dependent set, then one of them, say A, can be expressed as
a linear combination of the other two, A = uB + vC; this
equation says that the line segment representing A lies in the
same plane as the line segments that represent B and C. Thus,
if the three vectors form a linearly dependent set, their line
segments must be coplanar.
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Conversely, assume that A,B,C are vectors in R which
are represented by line segments drawn from the origin, all of
which lie in a plane. We claim that the vectors will then be
linearly dependent. To see this, let the equation of the plane
be ux + vy + wz = 0; keep in mind that the plane passes
through the origin. Let the entries of A be written ai, 02, 03,
with a similar notation for B and C. Then the respective
end points of the line segments have coordinates (01,02,03),
(61,62,63), (ci, 02,03). Since these points lie on the plane, we
have the equations

ua\ + vci2 + waz = 0
ub\ + v 62 + 1063 = 0
UO\ + VC2 + WC3 = 0

This homogeneous linear system has a non-trivial solution for
u, v, w, so the determinant of its coefficient matrix is zero by
3.3.2. Now the coefficient matrix of the linear system

' ua\ + vb\ + wci = 0
< ua 2 + vb 2 + WC2 = 0
k ^03 + 1*63 + wc 3 = 0

is the transpose of the previous one, so by 3.2.1 it has the same
determinant. It follows that the second linear system also has