Please answer C 3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which...
3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of a real variable). We have seen this vector space before, along with the usual operations of addition of functions and scalar multiplication of functions (see page 31-35 of the lecture notes). (ii) The real vector space C (R:R) (sometimes written as C(R is a vector subspace of F(R;R) that consists of real valued functions of a real variable that are differentiable infinitely many times at each point in R. Don't worry too much about the formality here: think of them as nice, smooth, well-behaved functions, eaning that you can differentiate, integrate as much as you want without any problems
3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of a real variable). We have seen this vector space before, along with the usual operations of addition of functions and scalar multiplication of functions (see page 31-35 of the lecture notes). (ii) The real vector space C (R:R) (sometimes written as C(R is a vector subspace of F(R;R) that consists of real valued functions of a real variable that are differentiable infinitely many times at each point in R. Don't worry too much about the formality here: think of them as nice, smooth, well-behaved functions, eaning that you can differentiate, integrate as much as you want without any problems