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In a previous lecture: Basis of the Null Space of a Matrix
This lecture: Column Space Basis
The column space of a matrix is defined in terms of a spanning set, namely the set of columns of the matrix. But the columns are not necessarily linearly independent. In this lecture, we demonstrate a systematic procedure for obtaining a linearly independent spanning set (i.e. a basis) for the column space of a matrix.
Consider the matrix A =
[
1 4 โ 1 6 4 5 0 โ 1 1 5 6 6 4 2 10 โ4 2 6 1 0 3 0 2 3
]
and also the matrix J =
[
1 00 1 (^) โ (^3) 1 0 (^0) โ (^21) โ (^31) 0 00 0 (^00 10 10 )
]
Notice that the column spaces of A and its โreducedโ matrix J are NOT the same. It is really easy to find a basis for the column space of a matrix in reduced eche- lon form, such as J.
Let the columns of J be called d 1 , ...d 6. Then by looking at the numerical entries in J, the following relationships can be seen:
d 3 = 3d 1 โ d 2 , d 5 = 2d 1 โ d 2 + d 4 , d 6 = 3d 1 โ d 2 + d 4
That is, d 3 , d 5 , d 6 are linear combinations of d 1 , d 2 , d 4. Columns 1, 2 and 4 are linearly independent so they form a basis of Col(J), the column space of J. Let the columns of matrix A be called c 1 , ...c 6. Now we can check that:
c 3 = 3c 1 โ c 2 , c 5 = 2c 1 โ c 2 + c 4 , c 6 = 3c 1 โ c 2 + c 4
[ โ 1 1 (^103)
]
[
1 (^04) 1
]
[
4 โ 1 (^20)
]
[
4 (^62) 2
]
[
1 (^04) 1
]
[
4 โ 1 (^20)
]
[
6 โ^54 0
]
[
5 (^66) 3
]
[
1 (^04) 1
]
[
4 โ 1 (^20)
]
[
6 โ^54 0
]
The reason the numbers add up this way is that the oper- ations of echelon reduction do not affect the dependency relations between the columns as the matrix A is trans- formed into J.
This leads to the matrix version of the famous Dimension Theorem of Vector Spaces.
Dimension Theorem for Matrices
Theorem If A is an m ร n matrix, then
dim Null(A) + dim Col(A) = number of columns of A
Proof Suppose that the reduced row echelon form of A has r leading 1โs. Then the left hand side of the above equation equals n โ r + r, which equals the number of columns of A.
Definition The nullity of a matrix A is the dimension of the Null Space of A.
Definition The rank of a matrix A is the dimension of the Column Space of A.
Therefore if A is an m ร n matrix whose reduced row echelon form J has r leading 1โs,
nullity = n โ r, rank = r and
rank + nullity = number of columns of the matrix A.
