Page 222 - 35Linear Algebra

222 Basis and Dimension

n
(Hint: You can build up a basis for B by choosing one vector at a
time, such that the vector you choose is not in the span of the previous
vectors you’ve chosen. How many vectors are in the span of any one
vector? Any two vectors? How many vectors are in the span of any k
vectors, for k ≤ n?)

Hint

3. Suppose that V is an n-dimensional vector space.

(a) Show that any n linearly independent vectors in V form a basis.
(Hint: Let {w 1 , . . . , w m } be a collection of n linearly independent
vectors in V , and let {v 1 , . . . , v n } be a basis for V . Apply the
method of Lemma 11.0.2 to these two sets of vectors.)

(b) Show that any set of n vectors in V which span V forms a basis
for V .

(Hint: Suppose that you have a set of n vectors which span V but
do not form a basis. What must be true about them? How could
you get a basis from this set? Use Corollary 11.0.3 to derive a
contradiction.)

4. Let S = {v 1 , . . . , v n } be a subset of a vector space V . Show that if every
vector w in V can be expressed uniquely as a linear combination of vec-
tors in S, then S is a basis of V . In other words: suppose that for every
n
1
vector w in V , there is exactly one set of constants c , . . . , c so that
n
1
c v 1 + · · · + c v n = w. Show that this means that the set S is linearly
independent and spans V . (This is the converse to theorem 11.0.1.)
5. Vectors are objects that you can add together; show that the set of all
3
linear transformations mapping R → R is itself a vector space. Find a
basis for this vector space. Do you think your proof could be modified
n
m
to work for linear transformations R → R? For R → R ? For R ?
N
R
3
Hint: Represent R as column vectors, and argue that a linear trans-
3
formation T : R → R is just a row vector.
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