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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
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