MAT225 Section Summary: 4.5

The Dimension of a Vector Space

Summary

Theorem 9: If a vector space V has a basis tex2html_wrap_inline203 , 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 tex2html_wrap_inline227 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

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Example: #11, p. 261

Theorem 12 (the Basis Theorem): Let V be a p-dimensional vector space, tex2html_wrap_inline247 . 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 tex2html_wrap_inline263 matrix. Then the dimension of Nul A is the number of free variables in the equation tex2html_wrap_inline267 , and the dimension of Col A is the number of pivot columns in A.

Example: #14, p. 261

Example: #27, 28, p. 262


LONG ANDREW E
Sat Jan 29 20:57:59 EST 2011