MATH 4377 Lecture Notes - Lecture 6: Natural Transformation, Linear Map, Coordinate Vector
Document Summary
Meeting 5 math 4377 and math 6308. Sections 2. 3 and 2. 4: invertible linear transformations, isomorphisms, change of coordinate matrix, similar matrices. Coordinate vector with respect to an ordered basis. Matrix representation for a linear transformation with respect to ordered bases. Theorem: suppose (cid:1848), (cid:1849) and (cid:1852) are finite dimensional vector spaces over (cid:1832). If (cid:1845):(cid:1848) (cid:1849) and (cid:1846):(cid:1849) (cid:1852) are linear transformations, then the composition (cid:1846) (cid:1845) is a linear transformation from (cid:1848) to (cid:1852). Theorem: suppose (cid:1848), (cid:1849) and (cid:1852) are finite dimensional vector spaces over (cid:1832), with bases , (cid:533) and (cid:534) respectively. If (cid:1845):(cid:1848) (cid:1849) and (cid:1846):(cid:1849) (cid:1852) are linear transformations, then (cid:4670)(cid:1846)(cid:1845)(cid:4671)(cid:3080)(cid:3082)(cid:3404)(cid:4670)(cid:1846)(cid:4671)(cid:3081)(cid:3082) (cid:4670)(cid:1845)(cid:4671)(cid:3080)(cid:3081) Theorem: suppose v and w are vector spaces over a field f, Definition: suppose v and w are vector spaces over a field f, and (cid:1846):(cid:1848) (cid:1849) is a linear transformation. Notation: (cid:1846)(cid:2879)(cid:2869) and (cid:1846):(cid:1848) (cid:1849) is a linear transformation. Furthermore, suppose there is a function (cid:1847):(cid:1849) (cid:1848) such that (cid:1847)(cid:1846)(cid:3404)(cid:1835)(cid:3023).
