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
BC
@yA = c1 @0A + c2 @1A + c3 @0A
z
0
0
1
has a solutio
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
combinations.
01
01
01
01
2
-1
2
-1
BC
BC
BC
BC
r1 (y1 @1A +
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 one real variable cfw_ f | f : R ! R has the subspace
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) Prove that for any four vectors ~ 1 , . . . , ~ 4 2 V we c
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 element of R2 as a linear combination of the two given ones.
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 space of
real-valued functions of one real variable?
(a)
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 subset.
1.12 Remark [Velleman] Denition 1.4 says that when
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 nothing special about that vector in
s
that equation. For any
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, the space
of linear polynomials cfw_ a + bx | a, b 2 R i
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 particular, notice that closure is a consequence
d2 (f
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 (1 - 2x + 3x2 - 4x3 ) - 2 (2 - 3x + x2 - (1/2)x3 ) = -
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 decides from the second equation that y = 1 and with that it
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 second row, etc.
for(row=1; row<=n-1; row+)cfw_
for(row_belo
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 computer. There are special purpose programs such as LIN
Topic
Analyzing Networks
The diagram below shows some of a cars electrical network. The battery is on
the left, drawn as stacked line segments. The wires are lines, shown straight and
with sharp right angles for neatness. Each light is a circle enclosing
Chapter One. Linear Systems
74
and also let the current through the battery be i0 . Note that we dont need to
know the actual direction of ow if current ows in the direction opposite to
our arrow then we will get a negative number in the solution.
" i0
i1
Section I. Denition of Vector Space
83
example of an addition in P
0
10101
1
-1
0
BCBCBC
1 A+@ 0 A=@ 1 A
@
-2
1
-1
illustrates that P is closed under addition. Weve added two vectors from P
that is, with the property that the sum of their three entries i
Section I. Denition of Vector Space
81
multiplication operations are always sensible they are dened for every pair of
vectors and every scalar and vector, and the result of the operation is a member
of the set (see Example 1.4).
1.3 Example The set R2 is
Chapter Two
Vector Spaces
The rst chapter nished with a fair understanding of how Gausss Method
solves a linear system. It systematically takes linear combinations of the rows.
Here we move to a general study of linear combinations.
We need a setting. At
Chapter One. Linear Systems
76
(c) This is a reasonably complicated network.
3 ohm
9 volt
3 ohm
3 ohm
2 ohm
4 ohm
2 ohm
2 ohm
2 In the rst network that we analyzed, with the three resistors in series, we just
added to get that they acted together like a s