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Why does row rank equal column rank?

3Column rank

In the previous example we saw that the linear equation

has a solution y=(y1,y2)TR2y=(y_1, y_2)^T\in\R^2 if and only if 2y1=y22 y_1 = y_2.

We can plot this to see that the yy’s which have a solution form a line in R2\R^2:

The elements for which the linear systems of equations have a solution form a line.

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

If y=(y1,y2)Ty=(y_1,y_2)^T and y=(y1,y2)Ty’=(y_1’, y_2’)^T are in the image of the coefficient matrix of the above linear system, then their sum y+yy+y’ is again in the image, as well as any multiple αy=(αy1,αy2)T\alpha\cdot y=(\alpha y_1, \alpha y_2)^T with αR\alpha\in\R.

This holds for the image of any matrix. We say that the image of a matrix is a subspace of Rm\R^m. It contains the columns of the matrix, since the ii-th column is AeiAe_i where ei=(0,,1,,0)Te_i = (0, \dots, 1, \dots, 0)^T with the 11 in the ii-th position. Moreover, one can show that it is the smallest subspace containing the columns. We say that im(A)\operatorname{im}(A) is spanned by the columns of A.

In particular, we can speak of the dimension of im(A)\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, im(A)\operatorname{im}(A) is one-dimensional.

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


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