MAT 1341 Lecture 12: Dimension theorems and bases

• School: University of Ottawa
• Department: Mathematics
• Course: MAT 1341
• Professor: Johannes Cuno

Document Summary

Every spanning set of v can be reduced to a basis of v. Every linearly independent subset of v can be extended to a basis of v: 2) start taking non-zero vectors from u forming larger and larger li sets or start with a [finite] spanning set of u and cut it down: dimension theorems. We know: the size of any linearly independent set in v is: If we know that v is finite dimensional and we want to find a basis for a subspace. U, we could either: and that a basis is the biggest possible li set in v and the smallest possible spanning set of v. Size of any spanning set of v: (cid:4666) (cid:4667) (cid:4666) (cid:4667) (cid:4666) (cid:4667, (cid:4666) (cid:4667) (cid:4666) (cid:4667) (cid:4666) (cid:4667) (cid:4666) (cid:4667) (cid:4666) (cid:4667) Then: any linearly independent set { is a basis of v any spanning set is a basis of v. If didn"t span v, pick v span{ .