Problem 6 Let op: VV be a linear map (also called "self-map" or endomorphism) of a vector space V. Given any real polynomial Pag+a+a2x²++ add, we can define the linear map P(): VV as aqldy +14+244 +...+ad popo...oy. d-times For any ≥0, we will denote pop0...04 = 4° (400 = Idy and 4°1 = 4). (-times As a reminder: for any polynomials P,Q € R[x], Q(p) P(4) P() Q() and (P+Q)() = P(x)+Q(4). = (QP)(x) = (PQ)(4) = (1) Given a polynomial P, show that for any E Ker(P()), () Є Ker(P()). hint: for a vector w being in the kernel of a linear map just means (w)=0+ try to show that and P(p) commute. So induces a linear self-map of the vector subspace Ker(P()). (2) Let ab be real numbers, we recall that 2- (a+b)x+ab = (-a)(x - b). We want to prove that Ker(p2 (a) Simplify (a+b)+ abldy) = Ker(-ald) Ker( - bld) ((-a) - (x - b)). (b) Apply the equality of polynomials obtained in the previous question to p. (c) Show that if Ker(2-(a+b)+abldy), then (-ald)(v) € Ker(p-bldy). hint: meaning of being in the kernel of a linear map? + use that polynomial multi- plication corresponds to composition, when applied to a linear map (d) Show that if Є Ker(p2-(a+b)+abldy), then (p-bld)(v) € Ker(p-aldy). (e) Now prove that Ker(2- (a + b) + abldy) = Ker(y-ald) + Ker( - bld). hint: double inclusion as usual. One inclusion uses (2) (b) and for the other, one can use the fact that if U, W, H are vector subspaces of V and UC H and WCH; then as H is closed under addition U+WCH.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.