Wednesday, February 10 − Lecture 17 : Subspaces associated to matrices
Concepts:
1. Nullspace.
2. Column-space
3. Row-space
4. Left Nullspace.
17.1 Definition − Let A m × nbe an m by n matrix. The nullspace, Null(A), of the matrix A
is defined as follows:
Null(A) = {x ∈ ℝ : Ax = 0}.
Determining the nullspace of a matrix A essentially comes down to solving a
homogeneous system of linear equations.
n m
If T : ℝ → ℝ is a linear mapping which induces the matrix A then
Ker(T) = Null(A).
- We know that the solution set S of a homogeneous system of linear equation is of the
form S = Span{a , a , …, a }. So the nullspace of the matrix A
1 2 m
Null(A) = Span{a , a , …, a }
1 2 m
is a subspace. By checking the linear independence of the spanning family {a 1, a2,
…, a m we can determine whether this spanning family is a basis of Null(A) or not.
17.2 Definition – Let A m×n = [c 1 … 2 ] n m×n be a matrix. Then the column space of A,
Col(A) is defined as
Col(A) = Span{c , 1 , 2, c } n
17.2.1 Theorem – Suppose where A = [c c … c 1 2 n m×n is an m by n. If T(x) = Ax,
then Range(T) = Col(A). Also, Col(A) = {b : Ax = b has a solution}
Proof : We have shown in the previous lecture (16.1.3) that Range(T) = Span{ c , c , 1 2
…, c n} = Col(A). Furthermore, b belongs to Span{c , c ,1…, 2 } if and only if b is a
linear combination of {c ,1c ,2…, c } nf and only if Ax = b has a solution.
17.2.2 Theorem – Suppose where A = [c c … c 1 2 n m×n is an m by n matrix with
Rank(A) = m. Then Col(A) = ℝ . m
Proof : We are given that Rank(A ) = m. Then A has a leading-1 in each row, and so
m×n RREF m
has no row of zeroes. Then Ax = b has a solution for all b in ℝ . That is b is linear
combination of { c 1 c 2 …, c }nfor all b. We conclude that Span{c , c1, 2, c } = n
m m
ℝ . That is Col(A) = ℝ .
17.2.3 Example – If A is the matrix
2
Then Col(A) = ℝ since A has rank equal to 2.
17.3 Definition – Let A be an m by n matrix with row vectors {r , r , …, r }. Then
m×n 1 2 m
the row space of A, Row(A), is defined as
Row(A) = Span{r , 1 , 2, r } m
17.3.1 Remarks – It follows immediately from the definition that the row space of
m
A m×n is a subspace of ℝ .
T T
Also note that given a matrix A m×n, Row(A) = Col(A ) and Row

More
Less