Let’s go back and consider a linear map . Remember that we defined its rank to be the dimension of its image. Let’s consider this a little more closely.
Any vector in the image of can be written as for some vector . If we pick a basis of , then we can write . Thus the vectors span the image of . And thus they contain a basis for the image.
More specifically, we can get a basis for the image by throwing out some of these vectors until those that remain are linearly independent. The number that remain must be the dimension of the image — the rank — and so must be independent of which vectors we throw out. Looking back at the maximality property of a basis, we can state a new characterization of the rank: it is the cardinality of the largest linearly independent subset of .
Now let’s consider in particular a linear transformation . Remember that these spaces of column vectors come with built-in bases and (respectively), and we have a matrix . For each index , then, we have the column vector
appearing as a column in the matrix .
So what is the rank of ? It’s the maximum number of linearly independent columns in the matrix of . This quantity we will call the “column rank” of the matrix.