Math 375 - 10/18

Theorem • V,W: vector spaces • x_1,…,x_n: basis for V • For any w_1,…,w_n∈W • There is a unique linear map T:V→W • s.t. {█(T(x_1 )=w_1@⋮@T(x_n )=w_n )┤ • v∈W⇒∃c_1,…,c_n∈R • s.t. v=c_1 x_1+…+c_n x_n • T(v)=c_1 w_1+…+c_n w_n • Linear map can be determined only by operations on basis Question 1 • Requirement ○ T: R2→R3 ○ dim⁡(range(T))=1 • Example ○ T(x,y)=(x,0,0) ○ T(x,y)=(0,y,y) ○ T(x,y)=(0,x+3y,−2x−6y) Question 2 • Requirement ○ T: R2→R2 ○ S: R2→R2 ○ ST=−TS • Example ○ T(x,y)=(−y,x) ○ S(x,y)=(−x,y) Question 3 • Requirement ○ T: R3→R3 ○ T^2≠0 ○ T^3≠0 • Example ○ T(x,y,z)=(0,x,y) Question 4 • Requirement ○ T: R2→R2 ○ T maps the unit square to the parallgram below ○ T(0,0)=(0,0) ○ T(1,0)=(2,0) ○ T(0,1)=(1,1) ○ T(1,1)=(3,0) • Example ○ T(x,y)=(2x+y,y) ○ T(x,y)=(2y+x,x) Question 5 • Requirement ○ T: R3→R3 ○ T(x,0,0)=(2x,0,0) ○ T^3 (0,a,b)=(0,a,b) • Example ○ T(x,y,z)=(2x,y,z) Question 6 • Requirement ○ T: R2→R2 ○ T(1,0)=(1,0) ○ {(x,y),T(x,y)} is independent whenever y≠0 • Example ○ T(x,y)=(x+y,y) Question 7 • Requirement ○ T: R2→R3 ○ T is injective ○ dim⁡(range(T))=1 • Example ○ Impossible ” t • — …. ice (1.1) T (1.1) (3.1) I _ _ _ _ _ ” - _ ~) > > (2.0)