3Column rank

In the previous example we saw that the linear equation

has a solution $y=(y_1, y_2)^T\in\R^2$ if and only if $2 y_1 = y_2$.

We can plot this to see that the $y$’s which have a solution form a line in $\R^2$:

We give these $y$’s, for which the linear system has a solution, a name: the image of the matrix. That is, for an $(m\times n)$-matrix $A$ we define

If $y=(y_1,y_2)^T$ and $y’=(y_1’, y_2’)^T$ are in the image of the coefficient matrix of the above linear system, then their sum $y+y’$ is again in the image, as well as any multiple $\alpha\cdot y=(\alpha y_1, \alpha y_2)^T$ with $\alpha\in\R$.

This holds for the image of any matrix. We say that the image of a matrix is a subspace of $\R^m$. It contains the columns of the matrix, since the $i$-th column is $Ae_i$ where $e_i = (0, \dots, 1, \dots, 0)^T$ with the $1$ in the $i$-th position. Moreover, one can show that it is the smallest subspace containing the columns. We say that $\operatorname{im}(A)$ is spanned by the columns of A.

In particular, we can speak of the dimension of $\operatorname{im}(A)$. Intuitively this is the number of different “directions” in the space. For example a single point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional, and a space looking like reality is three-dimensional. In our example, $\operatorname{im}(A)$ is one-dimensional.

The dimension of the image of a matrix has a special name: The column rank of the matrix.