Row and column spaces

Definition: Let A be n × m matrix. Then

• The row space, denoted row(A), of A is the subspace of ℝ^m given by the span of the rows of A.
• The column space, denoted col(A), of A is the subspace of ℝⁿ given by the span of the columns of A.

Theorem: Let A be a matrix and B an echelon form of A.

• The nonzero rows of B form a basis for row(A).
• The columns of A corresponding to the pivot columns of B form a basis for col(A).

Consequently, the dimension of the row space and the columns space of A are the same. We call this the rank of A, denoted rank(A).

Example: Let A be
$$\begin{bmatrix} 1 & 2 & 3 & 4 \\ 3 & -1 & 2 & 1 \\ 5 & 0 & 1 & -1 \end{bmatrix}$$
Find a basis for the row space. Find a basis for the column space. Determine the rank of A.

# We compute the rref of A and stare at it
A = matrix([[1,2,3,4],[3,-1,2,1],[5,0,1,-1]])
B = A.rref(); B
# The first 3 columns of A form a basis (so does the standard basis) for the column space
# The rows of B form a basis for the row space

[     1      0      0 -13/28]
[     0      1      0    1/4]
[     0      0      1  37/28]

Definition: The nullity of a matrix A, denoted null(A), is the dimension of the solution space to Ax = 0.

Example: What is the nullity of the previous A? (It is 1).

Theorem: (Rank-Nullity Theorem) Let A be a n × m matrix. Then rank(A) + nullity(A) = m.

Linear transform perspective

Let T : ℝⁿ → ℝ^m be a linear transform. Let A be the matrix so that T(x) = Ax. Then range(T) = col(A) so we know that hte rank of A is the dimension of the range. We know that the nullity is the dimension of the kernel. So dimension of range + dimension of kernel is the dimension of the domain.