Definition of column rank The definition of rank of a matrix is given as the dimension of the vector space spanned by it's columns. Is this the same as the number of linearly independent columns in the matrix? If so why isn't it defined (more simply and directly) as this?

They are two ways of looking at the same thing. If you have a matrix, it is more natural to think about the number of linearly independent columns. If you have a linear transformation, it is more natural to think about the dimension of the image. In different situations we start with one of these or the other, so it is useful to have both characterizations. For example, because we have the latter definition, we do not have to introduce a matrix representation in order to talk about the rank of a linear transformation. For another example, it is more apparent (to me, at least) that the latter definition does not depend on the bases for the domain and codomain that we use to build our matrix representation.
