| a c = 0 of the three-wide row vectors (b)
This subset of M22 cfw_ a b c d | a + d = 0 (c)
This subset of M22 cfw_ a b c d | 2a c d = 0
and a + 3b = 0 (d) The subset cfw_a + bx + cx3 | a
2b + c = 0 of P3 (e) The subset of P2 of
quadratic polynomials p s
a reader could suppose that our purpose in
this book is the study of linear systems. The
truth is that we will not so much use vector
spaces in the study of linear systems as we
instead have linear systems start us on the
study of vector spaces. The wide
if and only if r = 0. (b) Prove that r1 ~v = r2
~v if and only if r1 = r2. (c) Prove that any
nontrivial vector space is infinite. (d) Use the
fact that a nonempty solution set of a
homogeneous linear system is a vector space
to draw the conclusion. 1.41
functions of one natural number variable is a
vector space under the operations (f1 + f2) (n)
= f1(n) + f2(n) (r f) (n) = r f(n) so that if, for
example, f1(n) = n 2 + 2 sin(n) and f2(n) =
sin(n) + 0.5 then (f1 + 2f2) (n) = n 2 + 1. We can
view this spac
spanning sets that are minimal. Exercises X
2.20 Which of these subsets of the vector
space of 22 matrices are subspaces under the
inherited operations? For each one that is a
subspace, parametrize its description. For each
that is not, give a condition t
system in that a linear combination of these
solutions is also a solution. But it does not
contain all of the three-tall column vectors,
only some of them. We want the results about
linear combinations to apply anywhere that
linear combinations make sense
Example The definition requires that the
addition and scalar multiplication operations
must be the ones inherited from the larger
space. The set S = cfw_1 is a subset of R 1 . And,
under the operations 1 + 1 = 1 and r 1 = 1 the
set S is a vector space, sp
and sp(S). 2.14 Remark In Chapter One, after
we showed that we can write the solution set
of a homogeneous linear system as cfw_c1~ 1 +
+ ck~ k | c1, . . . , ck R, we described
that as the set generated by the ~ s. We
now call that the span of cfw_~ 1,
around in the left and right branches of the
parallel portion (we arbitrarily take the
direction of clockwise), there is a voltage rise
of 0 and a voltage drop of 8i2 12i1 so 8i2
12i1 = 0. i0 i1 i2 = 0 i0 + i1 + i2 = 0 12i1 =
20 8i2 = 20 12i1 + 8i2 = 0 T
parametrize it by expressing the leading
variable in terms of the free variables x = 2y
z. S = cfw_ 2y z y z | y, z R = cfw_y
2 1 0 + z 1 0 1 | y, z R ()
Now, to show that this is a subspace consider
r1~s1 + r2~s2. Each ~si is a linear combination
of th
conditions are just verifications. 1.20 For each,
list three elements and then show it is a vector
space. (a) The set of 22 matrices with real
entries under the usual matrix operations. (b)
The set of 22 matrices with real entries where
the 2, 1 entry is
scalar multipliction defined for this vector
space. The best way to understand the
definition is to go through the examples below
and for each, check all ten conditions. The first
example includes that check, written out at
length. Use it as a model for t
+ (w1 + u1) v2 + (w2 + u2) ! = v1 v2 ! + ( w1
w2 ! + u1 u2 ! ) For the fourth condition we
must produce a zero element the vector of
zeroes is it. v1 v2 ! + 0 0 ! = v1 v2 ! For (5), to
produce an additive inverse, note that for any
v1, v2 R we have v1 v2
cancellation of the integer parts on the right
side as well as on the left. (b) Solve the system
by hand, rounding to two decimal places, and
with = 0.001. Topic Analyzing Networks The
diagram below shows some of a cars electrical
network. The battery is
If S = cfw_~s1, . . . ,~sn is linearly independent
then S itself satisfies the statement, so assume
that it is linearly dependent. By the definition
of dependent, S contains a vector ~v1 that is a
linear combination of the others. Define the
set S1 = S c
dependent: cfw_ 1 0 0 , 3 0 0
independent: cfw_ 1 0 0 , 0 1 0
We got the dependent superset by adding
a vector from the x-axis and so the span did
not grow. We got the independent superset by
adding a vector that isnt in [S], because it has
a nonzero y
natural addition and scalar multiplication
operations. 1.26 Prove that this is not a vector
space: the set of two-tall column vectors with
real entries subject to these operations. x1 y1
+ x2 y2 = x1 x2 y1 y2 r x y = rx ry 1.27
Prove or disprove that R 3
integers (under the usual operations of
component-wise addition and scalar
multiplication). This is a subset of a vector
space but it is not itself a vector space. The
reason is that this set is not closed under
scalar 82 Chapter Two. Vector Spaces
multip
since traffic can flow through this network in
various ways; you should get at least one free
variable.) (b) Suppose that someone proposes
construction for Winooski Avenue East
between Willow and Jay, and traffic on that
block will be reduced. What is the
the same is only an intuition, but
nonetheless for each vector space identify the
k for which the space is the same as R k . (a)
The 23 matrices under the usual operations
(b) The nm matrices (under their usual
operations) (c) This set of 22 matrices cfw_
Example The set of solutions of a
homogeneous linear system in n variables is a
vector space under the operations inherited
from R n. For example, for closure under
addition consider a typical equation in that
system c1x1 + + cnxn = 0 and suppose that
bot
r x y z = rx ry rz The
addition and scalar multiplication operations
here are just the ones of R 3 , reused on its
subset P. We say that P inherits these
operations from R 3 . This Section I. Definition
of Vector Space 81 example of an addition in P
1 1
Example 1.5 gives a subset of R 2 that is not a
vector space, under the obvious operations,
because while it is closed under addition, it is
not closed under scalar multiplication.
Consider the set of vectors in the plane whose
components have the same si
1.10. Here, each member of the set has a finite
degree, that is, under the correspondence
there is no element from this space matching
(1, 2, 5, 10, . . .). Vectors in this space
correspond to infinite-tuples that end in
zeroes. 1.12 Example The set cfw_f
that we now know of: the trivial subspace,
lines through the origin, planes through the
origin, and the whole space. (Of course, the
picture shows only a few of the infinitely many
cases. Line segments connect subsets with
their supersets.) In the next se
subspaces, and we know that a good way to
understand a subspace is to parametrize its
description, we can try to understand a sets
span in that way. 2.18 Example Consider, in the
vector space of quadratic polynomials P2, the
span of the set S = cfw_3x x 2
a non-homogeneous linear system; do its
solutions form a subspace (under the inherited
operations)? 2.32 [Cleary] Give an example of
each or explain why it would be impossible to
do so. (a) A nonempty subset of M22 that is
not a subspace. (b) A set of two
The rows of this matrix A = 2 3 1 0 0 1 0
2 0 0 0 1 form a linearly independent
set. This is easy to check for this case but also
recall that Lemma One.III.2.5 shows that the
rows of any echelon form matrix form a
linearly independent set. 1.9 Example In
element of S, and it is obviously the additive
inverse of ~s under the inherited operations.
More information on equivalence of
statements is in the appendix. Section I.
Definition of Vector Space 93 The verifications
for the scalar multiplication conditi