MAT225 Section Summary: 4.2

Null spaces, column spaces, and linear transformations

Summary

The solution set of the homogeneous equation forms a subspace of , as one can see easily:

1. the zero vector is in the solution set (the trivial solution);
2. Consider two vectors in the solution set, and : then , so the solution set is closed under addition.
3. Consider a vectors in the solution set, and an arbitrary constant c: then , so the solution set is closed under scalar multiplication.

Null space of an matrix A: the null space of an matrix A, denoted Nul A, is the solution set of the homogeneous equation . It is the set of all that are mapped to the zero vector of by the transformation .

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

Example: #3, p. 234.

Notice that the number of vectors in the spanning set for Nul A equals the number of free variables in the equation .

Column space: Another subspace associated with the matrix A is the column space, Col A, defined as the span of the columns of A: . As a span, it is clearly a subspace (Theorem 3).

Col , which says that Col A is the range of the transformation .

Example: #16, p. 234

The null space lives in the row space of the matrix A, and the column space lives in the column space of A.

Example: #22, p. 235

Linear Transformation: A linear transformation T from a vector space V into a vector space W is a rule that assigns to each vector in V a unique vector in W, such that

1. T(u + v) = T(u) + T(v)
2. T(cu) = cT(u)

The kernel (or null space) of T is the set of such that . The range of T is the set of all vectors in W of the form for some in V.

Example: #30, p. 235

Examples of linear transformations include matrix transformations, as well as differentiation in the vector space of differentiable functions defined on an interval (a,b).

Example: #33, p. 235