4.3-row_and_column_spaces.md

Row and column spaces

Definition: Let $A$ be $n\times m$ matrix. Then

• The row space, denoted $row(A)$, of $A$ is the subspace of $\mathbb{R}^m$ given by the span of the rows of $A$.

• The column space, denoted $col(A)$, of $A$ is the subspace of $\mathbb{R}^n$ 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 $n\times m$ matrix. Then $rank(A)+nullity(A)=m$.

Linear transform perspective

Let $T:\mathbb{R}^n \to \mathbb{R}^m$ 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.