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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

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