MIT 18.06 - Lecture 10

Four subspaces

Any $m \times n$ matrix $A$ determines four subspaces (possibly containing only the zero vector):

Column space, $C(A)$

$C(A)$ consists of all combinations of the columns of $A$ and is a vector space in $R_m$.

The $r$ pivot columns form a basis for $C(A)$.

$$
dim \ C(A) = r.
$$

Nullspace, $N(A)$

$N(A)$ consists of all solutions $x$ of the equation $Ax = 0$ and lies in $R_n$.

The special solutions to $A_x = 0$ correspond to free variables and form a basis for $N(A)$. An $m \times n$ matrix has $n − r$ free variables:

$$
dim \ N(A) = n - r.
$$

Row space, $C(A ^ T)$

$C(A ^ T)$ consists of all combinations of the rows of $A$ and is a vector space in $R_n$.
We equate this with $C(A ^ T)$, the column space of the transpose of $A$.

To get the basis of the row space, one obvious way is to perform row reduction on $A ^ T$, but instead we make use of $R$, the row reduced echelon form of $A$.

During the process, it leads to the difference of column space between $A$ and $R$ (after many row changes operations).
Although the column spaces of $A$ and $R$ are different, the row space of $R$ is the same as the row space of $A$(row changes don’t influence the row space).

The rows of $R$ are combinations of the rows of $A$.

The first $r$ rows of $R$ are the ”echelon” basis for the row space of $A$:

$$
dim \ C(A ^ T) = r.
$$

Left nullspace, $N(A ^ T)$

We call the nullspace of $A ^ T$ the left nullspace of A. This is a subspace of $R_m$.

The matrix $A ^ T$ has $m$ columns. We just saw that $r$ is the rank of $A ^ T$, so the number of free columns of $A ^ T$ must be $m − r$.

$$
dim \ N(A ^ T) = m - r.
$$

The left nullspace is the collection of vectors $y$ for which $A ^ T y = 0.$ Equivalently, $y ^ T A = 0$; here $y$ and $0$ are row vectors.