Page 126 - A Course in Linear Algebra with Applications

110 Chapter Four: Introduction to Vector Spaces

which is not identically equal to zero in [0, 1].

Exercises 4.3

1. In each of the following cases determine if the subset S of
the vector space V is linearly dependent or linearly indepen-
dent:
(a) V = C and S consists of the column vectors

U)'KH'U4r) ;

(b) V = P(R) and S = {x - 1, x 2 + 1, x 3 - x 2 - x + 3};
(c) V = M(2, R) and S consists of the matrices

(2 - 3 \ / 3 1\ [12 -7\
\6 4J> 1,-1/2 - 3 / ' Vl7 6J'

2. A subset of a vector space that contains the zero vector is
linearly dependent: true or false?
3. If X is a linearly independent subset of a vector space, every
non-empty subset of X is also linearly independent: true or
false?

4. If X is a linearly dependent subset of a vector space, every
non-empty subset of X is also linearly dependent: true or
false?
5. Prove that any three vectors in R 2 are linearly dependent.
n
Generalize this result to R .
n
6. Find a set of n linearly independent vectors in R .
7. Find a set of ran linearly independent vectors in the vector
space M m>n (R).

8. Show that the functions x, e x sin x, e x cos x form a lin-
early independent subset of the vector space C[0, n].