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