Question

Let \(A\) be an \(m \times n\) matrix. We can view the columns of \(A\) as vectors, say \(a_1, \dots,, a_n\). The space of all linear combinations of the \(a_i\) are the column space of the matrix \(A\). Now, if \(a_1, \dots ,a_n\) are linearly independent, then the column space is of dimension \(n\). Otherwise, the dimension of the column space is the size of the maximal set of linearly independent \(a_i\). Row space is exactly analogous, but the vectors are the rows of \(A\).

The rank of a matrix A is the dimension of its column space - and - the dimension of its row space. These are equal for any matrix. Rank can be thought of as a measure of non-degeneracy of a system of linear equations, in that it is the dimension of the image of the linear transformation determined by \(A\).

The kernel of a matrix A is the dimension of the space mapped to zero under the linear transformation that \(A\) represents. The dimension of the kernel of a linear transformation is called the nullity.

Index theorem: For an \(m\times n\) matrix \(A\),

rank(\(A\)) + nullity(\(A\)) = \(n\).