Browse Section 4.2: Linear Transformations from Rn
to Rm.
Note the definitions for: linear transformation from
Rn to Rm, domainandcodomain for such a linear transformation,
standard matrix for such a transformation, and the
definition for a linear operatoron
Rn.
For one example, study the material on reflection
operators (bottom of p176 through middle of p177 plus
Tables 2,3 on pp177-178).
Study the remainder of Section 5.2.: Subspaces. Pay particular
attention to:
Definition (p215): w is a linear combination of
v1, v2, ..., vn
iff there exist scalars k1, k2,
...,kn, such that w =
k1v1
+k2v2
+...+knvn.
Theorem 5.2.3 p217: For v1,
v2, ..., vn vectors in some
vector space V, the collection of all linear combinations of
v1, v2, ...,
vn forms a subspace of V. Moreover, it is the
smallest subspace containing each of the vectors
v1, v2, ..., vn
.
Name for this subspace: (sub)space spanned by
v1, v2, ...,
vn.
Notation for this subspace: span{v1,
v2, ..., vn}.
Definition (and theorem): the vectors v1,
v2, ..., vn span the vector
space W iff each and every vector of W is a linear combination
of the vectors v1, v2, ...,
vn (thus iff W = span {v1,
v2, ..., vn}).
Example 12 p218: Showing three given vectors do not span
R3 by showing that a related matrix has
determinant zero. (So how could Long Theorem be expanded yet
again?)
Theorem 5.2.4 p219: Two sets of (finitely many) vectors in
V have the same span (or span the same subspace of V) iff each
vector in the first set is a linear combination of those in the
second and vice versa.