Direct Sum: Span of Basis Vectors

Question

We were asked to prove that the direct sum of the Column space of a matrix A, and the null space of the transpose of a matrix A , is R^3. My solution was to get the basis of both subspaces and show that the basis vectors spanned all of R^3. I am wondering if this makes sense, and if not, why not.

Answer 1

$$\dim(Null(A^T))=3-\dim(\mathcal{C}(A^T))=3-\dim(\mathcal{R}(A))=3-\dim(\mathcal{C}(A))$$ the last equality following from the fact that row rank is same as the column rank of a matrix. Refer this proof