MATH-M 303 Lecture Notes - Lecture 7: Linear Map, Scalar Multiplication
Document Summary
M303 section 1. 8 notes- introduction to linear maps/transformations. Ex. notion of a function knowledge from prior sections: because of relation to linear systems, we can answer many questions about matrix maps with. If is (cid:1865) (cid:1866) matrix, then for any vector (cid:3041), multiplication by produces new vector (cid:3040); if we regard vectors in (cid:3040) as inputs on which acts by multiplication to give output in (cid:3040), and we arrive at. Function/map (cid:1846): (cid:3041) (cid:3040)- rule which assigns unique output (cid:1846)(cid:4666)(cid:4667) (cid:3040) to each input (cid:3041) goes from domain (cid:3041) to target (cid:3040) Image of under (cid:1846)- output vector (cid:1846)(cid:4666)(cid:4667: range of (cid:1846)- set of all images of vectors {(cid:1846)(cid:4666)(cid:4667): (cid:3041)}; everything that gets hit by (cid:1846) Any (cid:1865) (cid:1866) matrix gives matrix map (cid:1846): (cid:3041) (cid:3040) given by (cid:1846)(cid:4666)(cid:4667)= If =[(cid:2183)(cid:2778) (cid:2183)(cid:2779) (cid:2183)(cid:2196)], then (cid:1846)(cid:4666)(cid:4667)==(cid:2869)(cid:2183)(cid:2778)+(cid:2869)(cid:2183)(cid:2779)+ +(cid:3041)(cid:2183)(cid:2196); range of (cid:1846) is set of linear combinations of columns of , which is (cid:1845)(cid:1866) {(cid:2183)(cid:2778),(cid:2183)(cid:2779), ,(cid:2183)(cid:2196)}=(cid:2869)(cid:2183)(cid:2778)+(cid:2869)(cid:2183)(cid:2779)+ +(cid:3041)(cid:2183)(cid:2196)
