# Math 20820: Homework 1 - University Of Notre Dame

Math 20820: Homework 1
Due Wednesday, Jan. 20.
Change of Basis: Let V be a ﬁnite dimensional vector space over a ﬁeld F with
dim V = n. Let
= (b
, b
, . . . , b ) and
= (c
, c
, . . . , c ) be two basis for V .
1
2
1
2
Then we have
I : V
V
= [ ]
= [ ]
(I) =
: F
F
And
=
[b
]
[b
]
[ b ]
.
1
2
That is the jth column of
is the coordinate vector for b in the
basis.
Then
is called the change of basis matrix from basis
to basis , and then
for T
(V ), we have
[T ]
= [T ] .
In particular, if
denotes the standard basis for F , then the change of basis
matrix on F from the
basis to the standard basis is given by
= [b
b
b ],
1
2
and the change of basis matrix from the
basis to the
basis is given by
1
1
=
=
= [c
c
. . . , c ]
[b
b
b ].
1
2
1
2
Deﬁnitions:
The trace of a matrix is deﬁned to be the sum of the terms on the diagonal of
the matrix.
a b
The determinant of a 2 x 2 matrix
is deﬁned to be the scalar ad
bc.
c d
A matrix is called a Markov matrix or a stochastic matrix if the entries of the
matrix are all real numbers in the interval [0, 1] and sum of the entries of each col-
umn of the matrix is 1.
An eigenvector with eigenvalue λ = 1 for a stochastic matrix is called a stable
equilibrium distribution.
1