Linear Algebra with Applications, 10th Edition

3.6 Row Space and Column Space

If A is an $m\times n$ matrix, each row of A is an n-tuple of real numbers and hence can be considered as a vector in ${ℝ}^{1\times n}$. The m vectors corresponding to the rows of A will be referred to as the row vectors of A. Similarly, each column of A can be considered as a vector in ${ℝ}^m$, and we can associate n column vectors with the matrix A.

Definition

If A is an $m\times n$ matrix, the subspace of ${ℝ}^{1\times n}$ spanned by the row vectors of A is called the row space of A. The subspace of ${ℝ}^m$ spanned by the column vectors of A is called the column space of A.

Example 1

Let

$$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$$

The row space of A is the set of all 3-tuples of the form

$$\alpha (1,0,0)+\beta (0,1,0)=(\alpha ,\beta ,0)$$

The column space of A is the set of all vectors of the form

$$\alpha \left[\begin{array}{c}1\\ 0\end{array}\right]$$