Column space and nullspace In this lecture we continue to study subspaces, particularly the column space and nullspace of a matrix. Review of subspaces A vector space is a collection of vectors which is closed under linear combina tions. In other words, for any two vectors v and w in the space and any two real numbers c and d, the vector cv + dw is also in the vector space. A subspace is a vector space contained inside a vector space. � � � � A plane P containing 0 0 0 and a line L containing 0 0 0 are both sub spaces of R3 . The union P ∪ L of those two subspaces is generally not a sub space, because the sum of a vector in P and a vector in L is probably not con tained in P ∪ L. The intersection S ∩ T of two subspaces S and T is a subspace. To prove this, use the fact that both S and T are closed under linear combina tions to show that their intersection is closed under linear combinations. Column space of A The column space of a matrix A is the vector space made up of all linear combi nations of the columns of A. Solving Ax = b Given a matrix A, for what vectors b does Ax = b have a solution x? ⎡ ⎤ 1 1 2 ⎢ 2 1 3 ⎥ ⎥ Let A = ⎢ ⎣ 3 1 4 ⎦. 4 1 5 Then Ax = b does not have a solution for every choice of b because solv ing Ax = b is equivalent to solving four linear equations in three unknowns. If there is a solution x to Ax = b, then b must be a linear combination of the columns of A. Only three columns cannot ﬁll the entire four dimensional vector space – some vectors b cannot be expressed as linear combinations of columns of A. Big question: what b’s allow Ax = b to be solved? A useful approach is to choose x and ﬁnd the vector b = Ax correspond ing to that solution. The components of x are just the coefﬁcients in a linear combination of columns of A. The system of linear equations Ax = b is solvable exactly when b is a vector in the column space of A. For our example matrix A, what can we say about the column space of A? Are the columns of A independent? In other words, does each column contribute something new to the subspace? The third column of A is the sum of the ﬁrst two columns, so does not add anything to the subspace. The column space of our matrix A is a two dimensional subspace of R4 . Nullspace of A � The nullspace of a matrix A is the collection of all solutions x = x1 x2 x3 � to the equation Ax = 0. The column space of the matrix in our example was a subspace of R4. The nullspace of A is a subspace of R3. To see that it’s a vector space, check that any sum or multiple of solutions to Ax = 0 is also a solution: A(x1 + x2) = Ax1 + Ax2 = 0 + 0 and A(cx) = cAx = c(0). In the example: ⎡ ⎤ ⎡ ⎤ ⎤ 1 1 2 ⎡ 0 x 1 ⎢ 2 1 3 ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ = ⎢ 0 ⎥, ⎣ 3 1 4 ⎦ x2 ⎣ 0 ⎦ x3 4 1 5 0 � the nullspace N ( A) consists of all multiples of 1 1 −1 � ; column 1 plus column 2 minus column 3 equals the zero vector. This nullspace is a line in R3 . Other values of b The solutions to the equation: ⎡ 1 1 ⎢ 2 1 ⎢ ⎣ 3 1 4 1 ⎤ ⎡ ⎤ 2 ⎡ 1 x1 ⎢ 2 3 ⎥ ⎥ ⎣ x2 ⎦ = ⎢ ⎣ 3 4 ⎦ x3 5 4 ⎤ ⎥ ⎥ ⎦ do not form a subspace. The zero vector is not a solution to this equation. The � � set of solutions forms a line in R3 that passes through the points � 0 −1 1 � � but not 0 0 0 � . 1 0 0 and