Sec.
5.2 Theory of Vector Spaces Associated with Systems of Equations
413
23. Give a constraint equation, if one exists, on the probability vectors in
the range of the transition matrices in Exercise 15.
24. By looking at the transition matrix for the frog Markov chain, explain
why the evenstate probabilities in the next period must equal the odd
state probabilities in the next period.
Hint:
Show that this equality is true if we start (this period) from a
specific state.
25. Compute the constraint on the range of the following powers of the frog
transition matrix. You will need a matrix software package.
(a) A2
(b) A3
(c) A!O
26. Show that the range
R(A)
of a matrix is a vector space. That is, if b
and b' are in
(for some x and x', Ax
=
b and Ax'
=
b'), show
that
rb
+
sb'
is in
R(A),
for any scalars
r, s.
27. Using matrix algebra, show that if x! and x
2
are solutions to the matrix
equation Ax
b, then any linear combination x'
=
ex!
+
dx
2
, with
e
+
d
=
.
1, is also a solution.
28. Show that the intersection
VI
n
V
2
of two vector spaces
V!, V
2
is again
a vector space.
Theory
of
Vector Spaces
Associated with Systems
Equations
In this section we introduce basic concepts abouJ vector spaces and use them
to obtain important information about the rangeand null space of a matrix.
Recall that a vector space
V
is a collection of vectors such that if u, v
E
V,
then any linear combination ru
+
sv is in
V.
In Section 5.1 we introduced
the range and null space of a matrix A:
Range(A)
=
{b : Ax
Null(A)
=
{x : Ax
b for some x}
O}
We noted that Range(A) and Null(A) are both vector spaces.
In Examples 2, 3, and 4 of Section 5.1 we used the elimination process
to find a vector or pair of vectors that generated the null spaces of certain
matrices. For example, multiples of [
1, 2,  2, 2,  2, 1] formed the
null space of the frog Markov transition matrix. In Examples 7, 8, and 9 of
Section 5.1, we used elimination to find constraint equations that vectors in
the range must satisfy. For the frog Markov matrix, the constraint for range
vectors p was
PI
+
P3
+
Ps
=
P2
+
P4
+
P6'
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Ch.
5 Theory of Systems of Linear Equations and EigenvaluelEigenvector Problems
The number of vectors generating the null space and number of con
straint equations for the range were dependent on how many pivots we made
during elimination. Our goal in this section is to show that the sizes of
Null(A) and Range(A) are independent of how elimination is performed.
The vector space
V
generated by a set
Q
=
{q!, q2,
... , q,} of
vectors is the collection of all vectors that can be expressed as a linear
combination of the q;'s. That is,
V
=
{v: v
For example, if
Q
consists of the unit nvectors e
j
(with all O's except for a
1 in positionj), then
V
is the vector space of all nvectors, that is, euclidean
nspace. Another name for a generating set is a
spanning set.
The column space of A, denoted Col (A), is the vector space generated
by the column vectors
af
of A. When we write
Ax =
b as
we see that the system
Ax
b has a solution if and only if b can be
expressed as a linear combination of the column vectors of A, or
Lemma
1. The system Ax
=
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 Fall '08
 FRIED
 Linear Algebra, Vector Space, Linear combination, Theory of Systems of Linear Equations

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