250 Lecture Index
We have already defined the column space and the null space of an n x m matrix to be the subspaces of Rn spanned by the columns and the subspace of solutions to the homogeneous system Ax = 0, respectively.
Example We consider a 4 x 5 matrix which (in row echelon form) has 3 pivots. The dimension of the column space is 3, and the dimension of the null space is 2.
This example is completely typical, in terms of the dimensions of these subspaces and the rank and nullity of the matrix.
Theorem The dimension of the column space of a matrix is the rank of the matrix. The dimension of the nullspace of a matrix is the nullity of the matrix.
Indeed, the rank counts the number of pivots in row echelon form, which gives the number of linearly independent vectors which spanthe column space. And the nullity counts the number of free variables in the solution to the homogeneous system, which is a count of the number of linearly independent vectors which span the null space.
Examples
What about the row space of a matrix? In fact, finding a basis for the row space turns out to be the easiest of the three under consideration, and this procedure will allow us to find its dimension. The main point to notice is that row operations do not change the span of the rows in question. It follows that the rows of a matrix in row echelon form have the same span as the rows in the original matrix from which this row echelon form was derived. But the nonzero rows of a matrix in row echelon form are linearly independent. Thus we see the following result.
Theorem
The nonzero rows of the row echelon form of a matrix form a basis for the
row space of the matrix.
The dimension of the row space of a
matrix is the rank of the matrix.
Examples
Notice that the rank of a matrix is the same as the dimension of the column or row space. Since transposition interchanges rows and columns, we can conclude that the rank of a matrix is unchanged under transposition.
Theorem A matrix and its transpose have the same rank.
Finally, let us consider the dimension properties of a pair of subspaces, with one subspace contained in the other.
Theorem Let V and W be subspaces of R n with V contained in W. Then dimV < dimW. Moreover, if dimV = dimW, then V= W.
Indeed, if we start with any basis for V, then it is a linearly independent subset of W, and therefore it can be expanded to a basis for W by the extension theorem. Therefore the number of basis elements in V is less than or equal to the dimension of W. If those numbers are equal, then the basis of V is a linearly independent subset of W with the same number of elements as the dimension of W, and so by the earlier theorem is a basis for W. In that case, it follows that V = W.