Editing Row and column spaces (section)

===Example===
Given a matrix {{mvar|J}}:
:<math>
  J =
  \begin{bmatrix}
    2 & 4 & 1 & 3 & 2\\
    -1 & -2 & 1 & 0 & 5\\
    1 & 6 & 2 & 2 & 2\\
    3 & 6 & 2 & 5 & 1
  \end{bmatrix}
</math>
the rows are
<math>\mathbf{r}_1 = \begin{bmatrix} 2 & 4 & 1 & 3 & 2 \end{bmatrix}</math>,
<math>\mathbf{r}_2 = \begin{bmatrix} -1 & -2 & 1 & 0 & 5 \end{bmatrix}</math>,
<math>\mathbf{r}_3 = \begin{bmatrix} 1 & 6 & 2 & 2 & 2 \end{bmatrix}</math>,
<math>\mathbf{r}_4 = \begin{bmatrix} 3 & 6 & 2 & 5 & 1 \end{bmatrix}</math>.
Consequently, the row space of {{mvar|J}} is the subspace of <math>\R^5</math> [[linear span|spanned]] by {{math|{{mset| '''r'''<sub>1</sub>, '''r'''<sub>2</sub>, '''r'''<sub>3</sub>, '''r'''<sub>4</sub> }}}}.   
Since these four row vectors are [[Linear independence|linearly independent]], the row space is 4-dimensional.  Moreover, in this case it can be seen that they are all [[orthogonality|orthogonal]] to the vector {{math|1='''n''' = [6, −1, 4, −4, 0]}} ({{math|1='''n'''}} is an element of the [[Kernel (linear algebra)|kernel]] of {{mvar|J}} ), so it can be deduced that the row space consists of all vectors in <math>\R^5</math> that are orthogonal to {{math|'''n'''}}.