Rank, Nullity, and Dimension

Row Rank
The row rank of a matrix is the dimension of the row space of the matrix. The row rank of a matrix is the maximum number of linearly independent rows in the matrix. Unsurprisingly, the row rank is denoted $rowrank(A)$.

Column Rank
The column rank of a matrix is the dimension of the column space of the matrix. The column rank of a matrix is the maximum number of linearly independent columns in the matrix. If you had to guess how the column rank is denoted, what would you say? $colrank(A)$? You're right!

If you didn't say $colrank(A)$...I don't know what to tell you.

Rank
The rank of a matrix is, interestingly, the same as the row rank and the column rank. There are a number of properties of the rank of a matrix:

• The system $A\vec{x}=\vec{b}$ is consistent if and only if $r(A)=r(\begin{bmatrix}A \,| \,\, b\end{bmatrix})$. This is clear because it means that the columns of $A$ with $b$ are linearly dependent, so $b$ can be expressed as a linear combination of $A$'s columns.
• An $n\times n$ matrix $A$ is invertible if and only if $r(A)=n$ ($A$ can be row-reduced to the identity matrix).
• If we have an consistent $n$-variable SLE $Ax=b$, then the solutions to the system are expressible in $n-r(A)$ unknowns (the number of zero rows after row-reducing).
• If $AB=I$, then the rows of $A$ are linearly independent, and $r(A)=n$.
• If $A$ is an $n\times n$ matrix such that $r(A)=n$, then there exists matrix $C$ such that $CA=I_n$.
• A square matrix is invertible if and only if it can be transformed to an upper triangular matrix with all diagonal elements equal to $1$ (no zero rows).
• $r(AB)\leq \min(r(A), r(B))$.

Column Space
Column space is the set of all possible outputs of $A\cdot\vec{v}$. This is just the span of the columns (the basis vectors)! For any $p\times n$ matrix $A$, $colrank(A)=rowrank(A)$. This means that the rank is equal to the column rank and the row rank. We can find bases for the column space in one of three ways:

1. By inspection.
2. By transposing, row-reducing, and taking nonzero columns.
3. By row-reducing (to $R$) and taking pivotal columns in $R$'s corresponding columns in $A$.

Row Space
The row space of a an $m\times n$ matrix $A$ is the span of the rows of $A$ each viewed as a matrix in $\mathbb{R}^n$. The row rank of a matrix is the dimension of the row space. The two are denoted $rowspace(A)$ and $rowrank(A)$. The basis of the row space of a matrix in row-reduced form is the nonzero rows of that matrix, and the row rank is the number of nonzero rows. Even better: if a matrix $B$ is obtained from matrix $A$ by applying some elementary row operations, then $rowspace(A)=rowspace(B)$. Also, if $B$ has some LI columns, the corresponding columns in $A$ are LI.

Full Rank
Not a poker hand! Full rank is when the rank of a transformation is the same as the number of input dimensions (number of columns).

Finding a Basis for the...

1. Row Space: Find the row-reduced form of the matrix. The basis is the set of nonzero rows.
2. Image/Column Space: Find the row-reduced form of the transpose of the matrix. The basis is the set of nonzero rows.
3. Kernel/Null Space: Find the row-reduced form of the matrix. The basis is the set of special solutions to the system $Ax=\textbf{0}$. Here, row-reducing isn't actually necessary, but it's helpful to solve $Ax=\textbf{0}$