124
Chapter Two. Vector Spaces
2.4 Lemma (Exchange Lemma) Assume that B = h~ 1 , . . . , ~ n i is a basis for a
vector space, and that for the vector ~ the relationship ~ = c1 ~ 1 + c2 ~ 2 + +
v
v
cn
138
Chapter Two. Vector Spaces
of the yz-plane; here are two such combinations.
01
01
01
01
0
1
0
1
0
w1
w1
0
w1
w1
BC
BC
BC
BC
B
C
B
C
@w2 A = 1 @w2 A + 1 @ 0 A
@w2 A = 1 @w2 /2A + 1 @w2 /2A
w3
0
w3
Chapter Two. Vector Spaces
140
and vice versa (we can move from the bottom to the top by taking each di to
be 1).
For (1) =) (2), assume that all decompositions are unique. We will show
_
_
that B1 Bk
Chapter Two. Vector Spaces
136
X
X
X
X
3.31 Show that the transpose operation is linear:
(rA + sB)T = rAT + sBT
for r, s 2 R and A, B 2 Mmn .
3.32 In this subsection we have shown that Gaussian reduct
Section III. Basis and Dimension
133
the columns containing the leading entries, h~ 1 , ~ 2 i. Thus, for a reduced echelon
ee
form matrix we can nd bases for the row and column spaces in essentially t
126
Chapter Two. Vector Spaces
2.13 Corollary Any linearly independent set can be expanded to make a basis.
Proof If a linearly independent set is not already a basis then it must not span
the space.
Section II. Linear Independence
115
(c) Find linearly independent sets S and T so that the union of S - (S \ T ) and
T - (S \ T ) is linearly independent, but the union S [ T is not linearly independe
142
Chapter Two. Vector Spaces
4.16 Example In R2 the x-axis and the y-axis are complements, that is, R2 =
x-axis y-axis. A space can have more than one pair of complementary subspaces;
another pair f
Section III. Basis and Dimension
III.3
129
Vector Spaces and Linear Systems
We will now reconsider linear systems and Gausss Method, aided by the tools
and terms of this chapter. We will make three po
144
Chapter Two. Vector Spaces
4.32 Recall that no linearly independent set contains the zero vector. Can an
independent set of subspaces contain the trivial subspace?
X 4.33 Does every subspace have
Section III. Basis and Dimension
117
1.4 Example The space R2 has many bases. Another one is this.
!
!
1
0
h
,
i
0
1
The verication is easy.
1.5 Denition For any Rn
0101
1
0
BCBC
B0C B1C
En = h B . C
Section II. Linear Independence
113
(b) f(x) = cos(x) and g(x) = sin(x)
(c) f(x) = ex and g(x) = ln(x)
X 1.23 Which of these subsets of the space of real-valued functions of one real variable
is linea
Topic
Fields
Computations involving only integers or only rational numbers are much easier
than those with real numbers. Could other algebraic structures, such as the
integers or the rationals, work i
Topic
Crystals
Everyone has noticed that table salt comes in little cubes.
This orderly outside arises from an orderly inside the way the atoms lie is
also cubical, these cubes stack in neat rows and
Section III. Basis and Dimension
119
terms, so that the two sums combine the same ~ s in the same order: ~ =
v
c1 ~ 1 + c2 ~ 2 + + cn ~ n and ~ = d1 ~ 1 + d2 ~ 2 + + dn ~ n . Now
v
c1 ~ 1 + c2 ~ 2 + +
Chapter Two. Vector Spaces
110
But there is no linearly independent superset os S. One way to see that is to
note that for any vector that we would add to S, the equation
01
01
01
01
x
1
0
0
BC
BC
BC
96
Chapter Two. Vector Spaces
Now, to show that this is a subspace consider r1~1 + r2~2 . Each ~i is a linear
s
s
s
combination of the two vectors in () so this is a linear combination of linear
combi
Chapter Two. Vector Spaces
94
2.6 Example Another example of a subspace that is not a subset of an Rn followed
the denition of a vector space. The space in Example 1.12 of all real-valued
functions of
Chapter Two. Vector Spaces
92
(b) This set
cfw_
0
b
a
| a, b 2 C and a + b = 0 + 0i
0
1.43 Name a property shared by all of the Rn s but not listed as a requirement for a
vector space.
X 1.44 (a) Pro
Chapter Two. Vector Spaces
98
with back substitution gives c2 = (x - y)/2 and c1 = (x + y)/2. This shows
that for any x, y there are appropriate coecients c1 , c2 making () true we
can write any eleme
Chapter Two. Vector Spaces
100
(b) x - x3 , cfw_ x2 , 2x + x2 , x + x3 , in P3
01
10
20
(c)
,cfw_
,
, in M22
42
11
23
2.23 Which of these are members of the span [cfw_ cos2 x, sin2 x ] in the vector s
108
Chapter Two. Vector Spaces
containing the zero vector has an element that is a combination of a subset of
other vectors from the set, specically, the zero vector is a combination of the
empty subs
Chapter Two. Vector Spaces
106
Observe that, although this way of writing one vector as a combination of
the others
~0 = c1~1 + c2~2 + + cn~n
s
s
s
s
visually sets o ~0 , algebraically there is nothin
Chapter Two. Vector Spaces
104
II
Linear Independence
The prior section shows how to understand a vector space as a span, as an
unrestricted linear combination of some of its elements. For example, th
Section I. Denition of Vector Space
87
1.14 Example The set
d2 f
+ f = 0
dx2
is a vector space under the, by now natural, interpretation.
cfw_f: R ! R |
(f + g) (x) = f(x) + g(x)
(r f) (x) = r f(x)
In
Section I. Denition of Vector Space
85
(the verication is easy). This vector space is worthy of attention because these
are the polynomial operations familiar from high school algebra. For instance,
3
Topic: Accuracy of Computations
69
The row reduction step -10001 + 2 produces a second equation -1001y =
-1000, which this computer rounds to two places as (-1.0 103 )y = -1.0 103 .
The computer decid
Topic
Accuracy of Computations
Gausss Method lends itself to computerization. The code below illustrates. It
operates on an n n matrix named a, doing row combinations using the rst
row, then the secon
Topic
Computer Algebra Systems
The linear systems in this chapter are small enough that their solution by hand
is easy. For large systems, including those involving thousands of equations,
we need a c