250 Lecture Index
We have considered the span of a set of vectors in Rn, and we know that this span is not necessarily all of Rn. On the other hand, it has important properties like Rn itself: linear combinations of elements in the span remain in the span. Such subsets of spaces of vectors are called subspaces, and we will consider how to relate such subspaces to other spaces Rm . (Geometrically, we have already seen that subspaces of R2 can look like {0} or lines through the origin, which look like R1, or R2 itself; similarly a subspace of R3 can be {0}, or a line through the origin, which looks like R1, or a plane through the origin, which look like R2, or else all of R3 itself.)
Definition
A set W of vectors in
Rn
is called a subspace of
Rn
if it has the following three
properties:
(1) 0
is in W;
(2) Whenever u and v belong to W, then
their sum u + v belongs to W ("closure under
addition");
(3) Whenever u
belongs to W and c is a scalar, then
the scalar multiple cu belongs to W
("closure under scalar multiplication").
Examples We consider sets given in a variety of fashions, including some for which either (2) or (3) holds, but not both.
We already know a way of generating subspaces of Rn.
Theorem The span of a nonempty subset of Rn is a subspace of Rn.
This theorem is just a restatement of properties we already know about the span, though we should note that to check property (1) we use the fact that 0 = 0v, and the span contains all sums of scalar multiples of elements of the nonempty subset.
Now, the span of the columns of an m x n matrix A is a subspace of Rm, which we have considered at length. It is called the column space of A, and will be denoted Col A. There are other important subspaces that are associated with a matrix, as well.
Definition
The null space of a matrix A is the set of all vectors
x satisfying Ax= 0. It is denoted
Null A.
Theorem The null space of an m x n matrix is a subspace of Rn.
Indeed, we have already seen that the three properties of being a subspace are satisfied by the collection of all solutions to a homogeneous space, so this theorem is just a restatement of results we already know. The null space can be considered as associated with the matrix A itself, or with the corresponding homogeneous equation Ax = 0.
Examples We consider spanning sets for some null spaces.
The rows of a matrix can also be considered as the spanning set of a space, just as the columns were.
Def. The row space of a matrix A is the set spanned by the rows of A. It is denoted Row A.