See rank, span, and kernel account for what a linear map keeps and loses.
Columns define the directions an output can use. A matrix combines its columns according to the input coordinates, so column independence controls the dimension of the image.
Two independent columns span both output dimensions.
Only the zero input maps to zero.
Input dimension = image dimension + lost dimension.
The two columns point in different directions, so their linear combinations fill the plane. Rank 2 means the image keeps both dimensions.
Rank measures how many independent output directions survive. Nullity measures how many input directions disappear. The rank–nullity theorem keeps the accounting exact: every lost input dimension appears in the kernel.