6.6 Column space, row space, and rank

This section is where matrix language and vector-space language meet directly. Once a matrix is fixed, two natural subspaces appear:

• the subspace generated by its columns;
• the subspace generated by its rows.

These are not cosmetic definitions. They answer real questions.

• Which right-hand sides b can occur in a system $Ax = b$?
• How much independent linear information is actually stored in the rows?
• How many genuinely independent directions remain after all redundancy is removed?

The answers are called the column space, the row space, and the rank of the matrix.

Column space: what outputs can the matrix produce?

Let $A$ be an $m \times n$ matrix, and write its columns as

$$A = \begin{bmatrix} a_1 & a_2 & \cdots & a_n \end{bmatrix}.$$

The most useful way to read this definition is through matrix multiplication. If $x = (x_1, \ldots, x_n)^t$, then

$$Ax = x_1 a_1 + x_2 a_2 + \cdots + x_n a_n.$$

So every vector of the form Ax is a linear combination of the columns. Conversely, every linear combination of the columns can be written as Ax for some vector x.

This is why column space matters in practice. It tells us precisely which right-hand sides are reachable.

Row space: what linear information do the rows carry?

If the rows of $A$ are $r_1, r_2, \ldots, r_m$, each regarded as row vectors in $R^n$, then we define a second subspace.

The ambient spaces are different in general:

• C(A) lives in $R^m$, because the columns each have m entries;
• R(A) lives in $R^n$, because the rows each have n entries.

Even when $A$ is square, the two spaces should not be confused. They are built from different vectors and answer different questions.

There is also a clean relation with transpose:

$$R(A) = C(A^t).$$

That identity lets you translate statements about row space into statements about column space of the transpose.

Row reduction: what changes and what does not?

Reduced row-echelon form is the main computational tool in this topic. But you have to use it carefully.

Why is this distinction so important?

• Row operations replace each new row by a linear combination of old rows, so the row space stays the same.
• Row operations act on whole rows, not on columns independently, so the actual column space usually changes.

This is the source of a standard rule that students often memorize before they understand it:

The pivot positions are read from the RREF, but the column-space basis vectors must be taken from the original matrix.

Rank counts the independent directions

The dimension of the column space is called the column rank. The dimension of the row space is called the row rank. One of the central theorems of linear algebra says that these numbers are equal.

When you row-reduce a matrix to RREF, the number of pivot columns is exactly the rank. So rank measures how many independent directions survive after all redundancy has been removed.

Worked example: read everything from one row reduction

Why the basis rule works

Suppose $R$ is the RREF of $A$.

• The pivot columns of $R$ show which column positions are independent.
• Row operations preserve linear relations among the columns.
• Therefore the corresponding columns of the original matrix $A$ are also independent and still span C(A).

For the row space, life is easier: row operations preserve the row space itself, so the nonzero rows of $R$ already form a basis of R(A).

Common mistake

Another common mistake is to say that column space and row space are equal because they have the same dimension. Equal dimension does not mean equal subspace; it only means they contain the same number of independent directions.

Quick checks

Exercises

Read this first

This note depends especially on 2.3 Gaussian elimination and RREF, 3.2 Transpose and special matrices, and 6.5 Basis and dimension.