This sections gives a new name to nullspace - kernel, and column space - range
The kernel of a transformation is the set of vectors that will be transformed to zeros
null space - all vectors in Rn that multiplication by A maps into 0; the null space is the solution space of the Ax = 0
so for linear transformations, the null space is the kernel
The range of a transformation is the set of possible outcomes
the column space in Ax = b spans 'b' - the column space spans all possible outcomes of Ax = b
so for linear transformations, the column space is range.
For completeness, here are the formal definitions in the book
| If T: V -> W is a linear transformation, then the set of vectors in V that T maps into 0 is called the kernel of T; it is denoted by ker(T). The set of all vectors in W that are images under T of at least one vector in V is called the range of T; it is denoted by R(T). |
| Example 2 Let T: V->W be the zero transformation (Example 2 of Section 8.1). Since T maps every vector in V into 0, it follows that ker(T) = V. Moreover, since 0 is the only image under T of vectors in V, we have R(T) = {0}. |
| Example 3 Let I: V -> V be the identity operator (Exmple 3 of Section 8.1). Since I(v) = v for all vectors in V, every vector in V is the image of some vector (namely itself); thus R(I) = V. Since the only vector that I maps into 0 is 0, it follows that ker(I) = {0}. |
| Example 6 - reworded If differentiation is considered to be a transformation, ker(T) = all constant functions - because the derivative of constant functions is zero |
The kernel and range are subspaces - they usually won't contain all vectors possible. This next theorem is a little obvious
| If T: V -> W is a linear transformation, then: a) The kernel of T is a subspace of V. b) The range of T is a subspace of W. |
As previously mentioned, under linear transformations, nullspace = the kernel, column space = the range, and so their dimensions must be the same. So here goes another obvious theorem
| If T: V -> W is a linear transformation, then: a) The kernel of T is a subspace of V. b) The range of T is a subspace of W. |
This next theorem is yet another carry over from chapter 4
| If T: V -> W is a linear transformation from an n-dimensional
vector space V to a vector space W, then rank(T) + nullity(T) = n |