Null Spaces, Column Spaces, and Linear Transformations Section 4.2 Null Spaces, Column Spaces, and Linear Transformations

In this section we will look at two important subspaces associated with a matrix, the null space and the column space.

Consider the following system of homogeneous equations: (1) Recall that in matrix form, this system may be written as where Remember that the set of all that satisfy (1) is called the solution set of the system. Often it is convenient to relate this solution set directly to the matrix A and the equation .

Definition The null space of an matrix A, written as Nul A, is the set of all solutions to Ax=0. Example: Find the null space of

Theorem 2 The null space of an matrix A is a subspace of . Proof:

Solving the equation produces an explicit description of Nul A. Ex: Find a spanning set for the null space of

For all problems of the previous type, the following is always true: The spanning set (produced using the method shown) is automatically linearly independent because the free variables are the weights of the spanning vectors. When Nul A contains the nonzero vectors, the number of vectors in the spanning set for Nul A equals the number of free variables in the equation Ax=0.

Definition The column space of an matrix A, written as Col A, is the set of all linear combinations of the columns of A. Note: Theorem 3 The column space of an matrix A is a subspace of .

Example: Find a matrix A such that W= Col A, where

Recall from Theorem 4 in Section 1 Recall from Theorem 4 in Section 1.4 that the columns of A span Rm if and only if the equation ___________ has a solution for _______________________ The column space of an ________ matrix A is all of ______ if and only if the equation __________ has a solution for each ____ in ______.

Example: 1. If the column space of A is a subspace of , what is k? 2. If the null space of A is a subspace of , what is k? 3. Find a nonzero vector in Col A and a nonzero vector in Nul A. 4. Determine if , are in Nul A and in Col A.

Definition: A linear transformation T from a vector space V into a vector space W is a rule assigns to each vector x in V a unique vector T(x) in W, such that (i) T(u+v)=T(u)+T(v) for all u, v in the domain of T: (ii) T(cu)=cT(u) for all u and all scalars c. If T(x)=Ax for some matrix A, Kernel of T = Nul A Range of T = Col A.