is a necessary and sufficient condition for a mapping to be a linear transformation. From this result we can conclude that a linear transformation
4.5.1 Invariant Subspaces
We adopt the notation L(V; W) to denote the set of linear transformations from the vector space V to the vector space W.
Definition 4.6 Let
We take some examples. Each subspace is invariant with respect to the following operators:
Let
We define the vector
Then P is a linear operator (called the projection operator) on to the subspace
The projection operator has the following invariant subspaces:
which remain unchanged and
that are carried into zero.
4.5.2 Rank and Nullity
Definition 4.7 Let
Definition 4.8 Let
Definition 4.9 The dimension of the subspace N(T) is called the nullity and is denoted by n(T).
For example: the zero operator ω has
Theorem 4.3 Let
We are now interested in determining in how far two vector spaces are ‘similar’ in some sense.
Theorem 4.4 Let
T is onto W (surjective) if and only if .
T is one-to-one (injective) if and only if .
Definition 4.10 A linear transformation
It
