, then any set in V containing more than n vectors
must be linearly dependent.
The Dimension of a Vector Space
Summary
Theorem 9: If a vector space V has a basis , then any set in V containing more than n vectors
must be linearly dependent.
Theorem 10: If a vector space V has a basis of n vectors, then every basis of V must consist of exactly n vectors.
dimension of a vector space: If V is spanned by a finite set, then V
is finite-dimensional, and the dimension V (dim V) is the number of
vectors in a basis for V. The dimension of the zero vector space
is defined to be zero. If V is not spanned by a finite set,
then V is said to be infinite-dimensional.
Example: #2, p. 260
Theorem 11 Let H be a subspace of a finite-dimensional vector space V. Any linearly independent set in H can be expanded, if necessary, to a basis for H. Also, H is finite-dimensional and
Example: #11, p. 261
Theorem 12 (the Basis Theorem): Let V be a p-dimensional vector
space, 
. Any linearly independent set of exactly p elements in V
is automatically a basis for V. Any set of exactly p elements that spans
V is automatically a basis for V.
Example: #22, p. 261
Let A be an 
matrix. Then the dimension of Nul A is the number of
free variables in the equation 
, and the dimension of Col
A is the number of pivot columns in A.
Example: #14, p. 261
Example: #27, 28, p. 262