5.4 Four fundamental subspaces

R(AT ) N(A).
N(A) R(AT ). Replacing A byAT N(AT ) R(A) .
Review of properties of orthogonal complements
Th2 If S is a subspace of Rn, then dim S + dim S = n. Furthermore, if {x1, ···, xr } is a basis for S and {xr+1, ···, xn} is a basis for S , then {x1, ···, xr , xr+1, ···, xn} is a basis for Rn.
Review of properties of orthogonal complements (continued)
Th3 If S is a subspace of Rn, then Rn = S ⊕S . From Th2
Th4 If S is a subspace of Rn, then (S ) S. From Th3
Four fundamental subspaces
Introduction
Four fundamental subspaces have important applications in many areas.
In this section we define four fundamental subspaces and show fundamental subspaces theorem and their orthogonalities.
Orthogonal decompositions of Euclidean vector spaces
If A is an m × n matrix, we have orthogonal decompositions of Euclidean vector spaces related to four fundamental subspaces.
spaces 4. Example
Definitions of four fundamental subspaces
If A is an m × n matrix, there are four fundamental subspaces:
row space of A in Rn : R(AT ) column space of A in Rm : R(A) null space of A in Rn : N(A) null space of AT in Rm : N(AT )
null space are orthogonal.
Next theorem tells us they are orthogonal complements of each other actually.
Fundamental subspaces theorem
Th1 If A is an m × n matrix, then
How are they related to each other?
Remark. Row space of a matrix and its null space are orthogonal.
R1 A x = 0,
A
R
2
,
Rix = (RiT) T x = 0, i =1, 2, 3.
Example
1 1 2
Ex.
Let
A=
0
1
1
.
1 3 4
Find the bases for N(A), R(AT ), N(AT ), and R(A). Solution.
R3 RiT and x are orthogonal, i =1, 2, 3.
Since any row vector of a matrix and any vector in its
null space are orthogonal, row space of a matrix and its
Proof of fundamental subspaces theorem
We have R(AT ) ⊥N(A). This implies that
N(A) R(AT ) . If x R(AT ) ,
then x is orthogonal to each of the column vectors of AT and, consequently, Ax = 0. Thus, x must be an element of N( A) and hence
Then we give the orthogonal decompositions of Euclidean vector spaces.
And we show an example to find the four fundamental subf four fundamental subspaces 2. Fundamental subspaces theorem 3. Orthogonal decompositions of Euclidean vector
Rn N(A) R(AT ),
By Th3
R(AT ) and N(A) are subspaces of Rn
Rm N(AT ) R(A).
R(A) and N(AT) are subspaces of Rm
How to find four fundamental subspaces? Finding bases
N(A) = R(AT ),
and N(AT) =R(A) .
Remark. Row space of a matrix and its null space are orthogonal complements. Column space of a matrix and the null space of its transpose are orthogonal complements.