MATH 4377 Lecture Notes - Lecture 5: Coordinate Vector, Linear Map
Document Summary
Meeting 4 math 4377 and math 6308. Sections 2. 2 and 2. 3: the matrix representation of a linear. Starting point: an ordered basis for a finite dimensional vector space v over a field f. Note: any basis becomes an ordered basis when we are very specific about the order of the vectors in the basis. Example: show that (cid:1828)={(cid:4672)(cid:883)(cid:883)(cid:4673),(cid:4672) (cid:883)(cid:883)(cid:4673)} is a basis for (cid:1844)(cid:2870). Then give the coordinate vector for (cid:4672)(cid:884)(cid:885)(cid:4673) relative to b. Example: we know (cid:1828)={(cid:4672)(cid:883)(cid:883)(cid:4673),(cid:4672) (cid:883)(cid:883)(cid:4673)} is a basis for (cid:1844)(cid:2870). Example: (cid:1828)={(cid:1876) (cid:883),(cid:1876)+(cid:883),(cid:1876)(cid:2870)+(cid:1876)} is a basis for (cid:2870)(cid:4666)(cid:1844)(cid:4667). and b is the standard ordered basis for (cid:2870)(cid:4666)(cid:1844)(cid:4667). Theorem: suppose v is a vector space over f, and (cid:1828)= {(cid:1874)(cid:2869), ,(cid:1874)(cid:3041)} is an ordered basis for v. then [ ] is a linear transformation from v to (cid:3041) that is one to one and onto. linear transformation, then the matrix representation of t in.
