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5. For t ER, define the evaluation map evt : Pn(R) + R given by evt(p(x))...
1. Let T : Pn(R) + Pn+1(R) be defined: T(P(x)) = (x + 1)p(x + 2) bases {1, X, ..., (a) (2 marks) Show that T is a linear transformation. (b) (3 marks) Is T one-to-one? Describe ker(T). What is the rank of T? (c) (8 marks) Find a matrix representation for T with respect to the standard xn} for Pn and {1, 2, ..., xn+1} for Pn+1 if n = 4. (d) (5 marks) Let D : Pn+1(R) +...
1. Let T : P (R) Pn+1(R) be defined: T(p()) = (x + 1)p(x + 2) (a) (2 marks) Show that T is a linear transformation. (b) (3 marks) Is T one-to-one? Describe ker(T). What is the rank of T? (c) (8 marks) Find a matrix representation for T with respect to the standard bases {1, 2, ..., 2"} for Pn and {1, 2, ..., xn+1} for Pn+1 if n = 4. (d) (5 marks) Let D : Pn+1(R) +...
1. Let T: Pn(R) + Pn+1(R) be defined: T(P(x)) = (x + 1)p(x + 2) (c) (8 marks) Find a matrix representation for T with respect to the standard bases {1, 2, ...,2"} for Pn and {1, 2, ...,xN+1} for Pn+1 if n = 4. (d) (5 marks) Let D : Pn+1(R) + Pn(R) be the derivative operator. What is the rank of DoT? Justify your answer. Describe ker(DoT). Is DoT one-to-one? (e) (5 marks) What is the rank of...
What's the solution of d and e 1. Let T : Pn(R) + Pn+1(R) be defined: T(P(x)) = (x + 1)p(x + 2) bases {1, X, ..., (a) (2 marks) Show that T is a linear transformation. (b) (3 marks) Is T one-to-one? Describe ker(T). What is the rank of T? (c) (8 marks) Find a matrix representation for T with respect to the standard xn} for Pn and {1, 2, ..., xn+1} for Pn+1 if n = 4. (d)...
1. Let 21,...,m ER be m distinct real numbers. Define m (p, q)m = p(x;) g(x3), j=1 for all p, q E P = {real polynomials}. Does (-;-)m define an inner product on P? If so, then prove it. If not, then give a counterexample. For which n e N does (-:-)m define an inner product on Pn = {p € P: deg p <n}. Make sure to justify your answer fully!
(*) Let D: Pn(R) + Pn-1(R) be a linear map with the property that for any non-constant polynomial p(x) € Pn(R), deg(D(p(x))) = deg(p(x)) – 1. Prove that D is surjective. Note: An example of such a D is the usual derivative function, but there are other possibilities as well!
2. Let ro < 1<..< n be n + 1 distinct points in IR. Define polynomials Co, ..., (n of degree n by (r - k) Let P, = 1,[r] be the polynomials of degree n, which is a vector space of dimension n + 1. (a) Show that the n+1 polynomials {lo, ..., Ln^ are basis for P i.e., they are linearly independent. (b) Find the coordinates [f]в of polynomial f E 1, with respect to the basis l-[10,...
Hi there, I literally got stuck on this question, it would be great if someone can give me help, many thanks in advance! A polynomial on R is a function p : R -R of the form p(x) - aj' where each ai E R and at most finitely many ai are nonzero. Let P denote the set of all polynomials on R (a) What is the dimension of P, regarded as a vector space over R? You do not...
3. Let U be a subspace of R", and let p :R" Rº be the projection v H proju v. Prove that a) p is a linear operator b) im T = U and ker T = U1 c) dim U + dim Ut = n.
1. Let T: R2 – R? be the map "reflection in the line y = x"—you may assume this T is linear, let Eº be the standard basis of R2 and let B be the basis given by B = a) On the graph below, draw a line (colored if possible) joining each of the points each of the points (-). (). (1) and () woits image to its image under the map T. y = x b) Find the...