Lecture 9 dim & rank - 4-5 & 4-6

© 2012 Pearson Education, Inc.

Vector Spaces

THE DIMENSION OF A

VECTOR SPACE

5- 2© 2012 Pearson Education, Inc.

DIMENSION OF A VECTOR SPACE

Theorem 9: If a vector space V has a basis

B = {b1, … ,bn} then any set in V containing more

than n vectors must be linearly dependent.



Theorem 10: If a vector space V has a basis of n

vectors, then every basis of V must consist of

exactly n vectors.



Definition: If V is spanned by a finite set, then V is

said to be finite-dimensional, and the dimension

of V, written as dim V, is the number of vectors in a

basis for V. The dimension of the zero vector

space {0} is defined to be zero. If V is not spanned

by a finite set, then V is said to be infinite-

dimensional.

Dimension - Examples

Dim Rn?

Dim Pn?

𝐯1 =362

, 𝐯2 =−101

, H=Span{v1,v2}, Dim

H?

Slide 2.2- 5© 2012 Pearson Education, Inc.

Dimension - Examples

Find the dimension of the subspace


3 6

5 4: , , , in

2

5

a b c

a dH a b c d

b c d

d

Geometric Interpretation of Subspaces of R3

0-dimensional origin

1-dimensional line through the origin

2-dimensional plane through the origin

3-dimensional all of R3



SUBSPACES OF A FINITE-DIMENSIONAL

SPACE

Theorem 11: Let H be a subspace of a finite-

dimensional vector space V. Any linearly

independent set in H can be expanded, if

necessary, to a basis for H. Also, H is finite-

dimensional and dim H ≤ dim V


THE BASIS THEOREM

Theorem 12: Let V be a p-dimensional vector

space,

p ≥ 1. Any linearly independent set of exactly p

elements in V is automatically a basis for V. Any

set of exactly p elements that spans V is

automatically a basis for V.


THE DIMENSIONS OF NUL A AND COL A

dim Nul(A) = # free vars in Ax=0

dim Col(A) = # pivot columns in A


DIMENSIONS OF NUL A AND COL A

Example 2: Find the dimensions of the null

space and the column space of

3 6 1 1 7

1 2 2 3 1

2 4 5 8 4

A

© 2012 Pearson Education, Inc.

Vector Spaces

RANK


THE ROW SPACE

If A is an matrix, each row of A has n

entries and thus can be identified with a vector in

Rn.

The set of all linear combinations of the row

vectors is called the row space of A and is

denoted by Row A.

Since the rows of A are identified with the

columns of AT, we could also write Col AT in

place of Row A.

m n


THE ROW SPACE

Theorem 13: If two matrices A and B are row

equivalent, then their row spaces are the same. If

B is in echelon form, the nonzero rows of B form a

basis for the row space of A as well as for that of

B.

Important:

Use pivot cols of A for basis of Col(A)

Use pivot rows of REF(A) for basis of Row(A)


THE ROW SPACE - Example

Example 1: Find bases for the row space, the

column space, and the null space of the matrix

2 5 8 0 17

1 3 5 1 5

3 11 19 7 1

1 7 13 5 3


RANK- Definition

Definition: rank of A is the dim Col(A).

Since Row A is the same as Col AT, the

dimension of the row space of A is the rank of

AT.

The dimension of the null space is sometimes

called the nullity of A.

THE RANK THEOREM

Theorem 14: The dimensions of the column

space and the row space of an mxn matrix A

are equal. This common dimension, the rank

of A, also equals the number of pivot

positions in A and satisfies the equation:

rank A + dim Nul(A) = n



THE RANK THEOREM

Proof:

dim Col(A) # pivot columns in REF(A) = #pivots

dim Row(A) # nonzero rows in REF(A) = #pivots

dim Nul(A) #free vars in Ax=0 = #nonpivot cols


THE RANK THEOREM - Example

Example 2:

a. If A is a 7x9 matrix with a two-dimensional null

space, what is the rank of A?

b. Could a 6x9 matrix have a two-dimensional null

space?


THE INVERTIBLE MATRIX THEOREM

(CONTD)

Theorem: Let A be an nxn matrix. Then the

following statements are each equivalent to the

statement that A is an invertible matrix.

m. The columns of A form a basis of Rn.

n. Col(A) = Rn

o. dim Col(A) = n

p. rank A = n

q. Nul(A) = {0}

r. dim Nul(A) = 0

Practice Problem

𝐴 =

2 −1 1 −6 81 −2 −4 3 −2−7 8 10 3 −104 −5 −7 0 4

~

1 −2 −4 3 60 3 9 −12 120 0 0 0 00 0 0 0 0

1. Find rank A

2. Find dim Nul(A)

3. Find basis for Col(A)

4. Find basis for Row(A)

5. What is the next step to find basis for Nul(A)?

6. How many pivots in REF(AT)?


Markov Chains


State 1 State 2

Probability going

from 1 to 2

Prob

staying

in state

1

Probability going

from 2 to 1

Prob

staying

in state

2

Stochastic Matrix

Square matrix

Columns are probability vectors

aij is probability of going from state i to state j


Markov Chain

Sequence of state vectors, xk, such that

xk+1 = Pxk

Next state depends only on current state


Markov Chains - Example


City Suburbs

.4

.6

.3

.7

Stochastic Matrix - Example

Find the stochastic matrix for the previous

example.

Use the stochastic matrix to predict next

year’s populations if this year’s

populations are:

City: 600,000

Suburbs: 400,000


Steady State Vector

Pq= q

Every stochastic matrix has a steady state

vector

Theorem 18: If P is an nxn regular stochastic

matrix, then P has a unique steady-state

vector q. Further, if x0 is any initial state, the

Markov chain converges to q.

Regular: some power of P contains only

strictly positive entriesSlide 2.2- 27© 2012 Pearson Education, Inc.

Regular Matrix - Example

Is P = 1 .30 .7

a regular stochastic matrix?


Steady State Vector - Example

Find the steady state vector for the

city/suburb example.


Lecture 9 dim & rank - 4-5 & 4-6

Education

Transcript of Lecture 9 dim & rank - 4-5 & 4-6