5.5 Row Space, Column Space, and Nullspace

2

Row Space, Column Space, and Nullspace Definition:

For an mxn matrix

the vectors

in Rn formed from the rows of A are called the row vectors of A, and the vectors

in Rm formed from the columns of A are called the column vectors of A

mnmm

n

n

aaa

aaa

aaa

A

21

22221

11211

mnmm

n

n

aaa

aaa

aaa

21

22221

11211

m

2

1

r

r

r

mn

m

m

n

nn a

a

a

c

a

a

a

c

a

a

a

c

2

1

2

22

21

2

1

12

11

1 ,,,

3

Row Space, Column Space, and Nullspace Definition:

If A is an mxn matrix, then the subspace of Rn spanned by the row vectors of A is called the row space of A,

and the subspace of Rm spanned by the column vectors is called the column space of A.

The solution space of the homogeneous system of equations Ax=0, which is a subspace of Rn, is called the nullspace of A.

Theorem 5.5.1: A system of linar equations Ax=b is consistent iff b is in the column space of A

4

example

1 3 2

1 2 3

2 1 2

1

2

3

x

x

x

1

9

3

Show that b in column space of A!

The solution by G.E. X1 = 2, X2 = -1, X3 = 3, the system is consistent, b is in the column space of A

1 3 2 1

2 1 2 3 3 9

1 1 2 3

5

Row Space, Column Space, and Nullspace Theorem 5.5.2: If x0 denotes any single solution

of a consistent linear system Ax=b, and if v1,v2,...,vk form a basis for the nullspace of A, that is, the solution space of the homogeneous system Ax=0, then every solution of Ax=b can be expressed in the form

and, conversely, for all choices of scalars c1,c2,...,ck, the vector x in this formula is a solution of Ax=b.

k210 vvvxx kccc 21

6

General and Particular Solutions

Terminology: Vector x0 is called a particularly solution of Ax=b.

The expression x0+c1v1+c2v2+...+ckvk is called the general solution of Ax=b.

The expression c1v1+c2v2+...+ckvk is called the general solution of Ax=0.

7

Bases for Row Spaces, Column Spaces, and Nullspaces Theorem 5.5.3: Elementary row operations do not

change the nullspace of a matrix Theorem 5.5.4: Elementary row operations do not

change the row space of a matrix Theorem 5.5.5: If A and B are row equivalent

matrices, then:a) A given set of column vectors of A is linearly

independent iff the corresponding column vectors of B are linearly independent.

b) A given set of column vectors of A forms a basis for the column space of A iff the corresponding column vectors of B form a basis for the column space of B.

8

Bases for Row Spaces, Column Spaces, and Nullspaces Theorem: If a matrix R is in row-echelon form, then the row

vectors with the leading 1’s (i.e., the nonzero row vectors) form a basis for the row space of R, and the column vectors with the leading 1’s of the row vectors form a basis for the column space of R.

Example: [Bases for Row and Column Spaces]The matrix R is in row-echelon form, while the vectors r

form a basis for the row space of R

00000

01000

00310

30521

R

01000

00310

30521

3

2

1

r

r

r

9

Bases for Row Spaces, Column Spaces, and Nullspaces

and the vectors

form a basis for the column space of R Example: [Bases for Row and Column Spaces]

Find bases for the row and column spaces of

0

1

0

0

,

0

0

1

2

,

0

0

0

1

4ccc 21

000000

510000

623100

452431

R operations row

elementary

452431

791962

281962

452431

A

10

Bases for Row Spaces, Column Spaces, and Nullspaces

The basis vectors are

The first, third, and fifth columns of R contain the leading 1’s of the row vectors that form a basis for the column space of R.

Thus the corresponding column vectors of A, form a basis for the column space of A

510000

623100

452431

3

2

1

r

r

r

0

1

2

5

,

0

0

1

4

,

0

0

0

1

531 ccc

5

9

8

5

,

4

9

9

4

,

1

2

2

1

531 ccc

11

Bases for Row Spaces, Column Spaces, and Nullspaces Example: [Basis and Linear Combinations]

a) Find a subset of the vectors

v1=(1,-2,0,3), v2=(2,-5,-3,6), v3=(0,1,3,0),

v4=(2,-1,4,-7), v5=(5,-8,1,2) that forms a basis for the space spanned by these vectors.

b) Express each vector not in the basis as a linear combination of the basis vector

Solution:

a)

0 0 0 0 0

1 1 0 0 0

1 0 11 0

1 0 2 0 1

27063

14330

81152

52021

5432154321 vvvvv wwwww

formechelon

row

reduced

12

Bases for Row Spaces, Column Spaces, and Nullspaces

Basis for the column space of matrix vectors w is {w1,w2,w4} and consequently basis for the column space of matrix vectors v is {v1,v2,v4}.

b) Expressing w3 and w5 as linear combinations of the basis vectors w1,w2, and w4 (dependency equations).

w3 = 2w1 - w2

w5 = w1 + w2 + w4

The corresponding relationships are

v3 = 2v1 – v2

v5 = v1 + v2 + v4

13

Bases for Row Spaces, Column Spaces, and Nullspaces Given a set of vectors S={v1,v2,...,vk) in Rn, the following

procedure produces a subset of these vectors that forms a basis for span(S) and expresses those vectors of S that are not in the basis as linear combinations of the basis vectors.

Step 1. Form the matrix A having v1,v2,...,vk as its column vectors.Step 2. Reduce the matrix A to its reduced row-echelon form R, and let w1,w2,...,wk be the column vectors of R.Step 3. Identify the columns that contain the leading 1’s in R. The corresponding column vectors of A are the basis vectors for span(S).Step 4. Express each column vector of R that does not contain a leading 1 as a linear combination of preceding column vectors that do contain leading 1’s.

5.6 Rank and Nullity

15

Four Fundamental Matrix Spaces

Fundamental matrix spaces:Row space of A, Column space of A

Nullspace of A, Nullspace of AT

Relationships between the dimensions of these four vector spaces.

16

Row and Column Spaces have Equal Dimensions Theorem 5.6.1: If A is any matrix, then the

row space and column space of A have the same dimension.

The common dimension of the row space and column space of a matrix A is called the rank of A and is denoted by rank(A); the dimension of the nullspace of A is called the nullity of A and is denoted by nullity(A).

17

Row and Column Spaces have Equal Dimensions Example: [Rank and Nullity of a 4x6 Matrix]

Find the rank and nullity

of the matrix

Solution:

The reduced row-echeclon

form of A is

rank(A) = 2 and the corresponding system will be

744294

164252

410273

354021

A

000000

000000

51612210

133728401

0516122

01337284

65432

65431

xxxxx

xxxxx

18

Row and Column Spaces Have Equal Dimensions

The general solution of the system is

65432

65431

516122

337284

xxxxx

xxxxx

ux

tx

sx

rx

utsrx

utsrx

6

5

4

3

2

1

516122

337284

19

Row and Column Spaces Have Equal Dimensions

Nullity(A)=4

1

0

0

0

5

13

0

1

0

0

16

37

0

0

1

0

12

28

0

0

0

1

2

4

6

5

4

3

2

1

utsr

x

x

x

x

x

x

20

Row and Column Spaces Have Equal Dimensions Theorem 5.6.2: If A is any matrix, then rank(A) =

rank(AT). Theorem 5.6.3: [Dimension Theorem for Matrices]

If A is a matrix with n columns, then

rank(A) + nullity(A) = n Theorem 5.6.4: If A is an mxn matrix, then:

a) Rank(A) = the number of leading variables in the solution of Ax = 0.

b) Nullity(A) = the number of parameters in the general solution of Ax = 0.

21

Row and Column Spaces Have Equal Dimensions A is an mxn matrix of rank r

Fundamental Space Dimension

Row space of A r

Column space of A r

Nullspace of A n-r

Nullspace of AT m-r

22

Maximum Value for Rank

A is an mxn matrix:

rank(A) ≤ min(m,n)

where min(m,n) denotes the smaller of the numbers m and n if m≠n or their common value if m=n.

23

Linear Systems of m Equations in n Unknowns Theorem 5.6.5: [The Consistency Theorem]

If Ax = b is a linear system of m equations in n unknowns, then the following are equivalent.

a) Ax = b is consistent

b) b is in the column space of A.

c) The coefficient matrix A and the augmented matrix [A|b] have the same rank.

24

Linear Systems of m Equations in n Unknowns Theorem:

If Ax = b is a linear system of m equations in n unknowns, then the following are equivalent.

a) Ax = b is consistent for every mx1 matrix b.

b) The column vectors of A span Rm.

c) Rank(A) = m A linear system with more equations than unknowns

is called an overdetermined linear system. The system cannot be consistent for every possible b.

25

Linear Systems of m Equations in n Unknowns Example: [Overdetermined System]

The system is consistent

iff b1, b2, b3, b4, and b5

satisfy the conditions

521

421

321

221

121

3

2

2

bxx

bxx

bxx

bxx

bxx

125

124

123

12

12

4500

3400

2300

10

201

bbb

bbb

bbb

bb

bb

arbitrary are and where

,,2

,34,45

054

043

032

543

21

521

421

321

sr

sbrbsrb

srbsrb

bbb

bbb

bbb

26

Linear Systems of m Equations in n Unknowns Theorem 5.6.7: If Ax=b is a consistent linear system of

m equations in n unknowns, and if A has rank r, then the general solution of the system contains n-r parameters.

Theorem 5.6.8: If A is an mxn matrix, then the following are equivalent.

a) Ax=0 has only the trivial solution.b) The column vectors of A are linearly independent.c) Ax=b has at most one solution (none or one) for every

mx1 matrix b. A linear system with more unknowns than equations is

called an underdetermined linear system. Underdetermined linear system is consistent if its

solution has at least one parameter → has infinitely many solution.

27

Summary

Theorem 5.6.9: [Equivalent Statements]

If A is an nxn matrix, and if TA:Rn→Rn is multiplication by A, then the following are equivalent.

a) A is invertible

b) Ax=0 has only the trivial solution

c) The reduced row-echelon form of A is In.

d) A is expressible as a product of elementary matrices.

e) Ax=b is consistent for every nx1 matrix b

f) Ax=b has exactly one solution for every nx1 matrix b

g) Det(A)≠0

28

Summary

h) The range of TA is Rn

i) TA is one-to-one

j) The column vectors of A are linearly independent

k) The row vectors of A are linearly independent

l) The column vectors of A span Rn

m) The row vectors of A span Rn

n) The column vectors of A form a basis for Rn

o) The row vectors of A form a basis for Rn

p) A has rank n

q) A has nullity 0