Page 122 - A Course in Linear Algebra with Applications

106 Chapter Four: Introduction to Vector Spaces

a non-trivial solution u, v, w. But then uA + vB + wC = 0,
which shows that the vectors A, B, C are linearly independent.
Thus there is a natural geometrical interpretation of lin-
3
ear dependence in the Euclidean space R : three vectors are
linearly dependent if and only if they are represented by line
segments lying in the same plane. There is a corresponding in-
terpretation of linear dependence in R 2 (see Exercise 4.3.11).
Example 4.3.1

Are the polynomials x + 1, x + 2, x 2 — 1 linearly dependent in
the vector space P 3 (R)?

To answer this, suppose that c\,C2,c^ are scalars satisfy-
ing
2
c x{x + 1) + c 2(x + 2)+ c 3(x - 1) = 0.
2
Equating to zero the coefficients of 1, x , x , we obtain the
homogeneous linear system

Ci + 2 c 2 - C3 = 0
ci + c 2 = 0
C 3 = 0
This has only the trivial solution c\ = c 2 = c 3 = 0; hence the
polynomials are linearly independent.
Example 4.3.2

Show that the vectors

( - : ) • ( ! ) • ( - : )

2
are linearly dependent in R .
Proceeding as in the last example, we let c\, c 2, cz be
scalars such that
+
*(-J) *GM-3-C0-