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 even-state 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'