1. (17 points) Consider the following set: S = = { [@PO], where p(t) = at...

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1. (17 points) Consider the following set: S = = { ["@PO], where p(t) = at² + bt + c. [p(0) p(0)] (A) Show that the set is non-empty. (solution) (B) Determine whether there exists the additive identity of the set or not. If yes, find it. Otherwise, explain why not. (solution) (C) Determine whether the set is closed under addition or not. Explain why or why not. (solution) (D) Determine whether the set is closed under scalar multiplication or not. Explain why or why not. (solution) (E) Determine whether the set is a vector space or not. Explain why or why not. (solution) (F)-(P) Let T: P2 → S be defined by T (p(t)) = PO PO, where P₂ = {at² + bt + c, where a, b and c are any real numbers.}, and S is the set defined above. (G) Determine whether T is linear or not. Explain why or why not. (solution) (H) Find the kernel of T. (solution) This is an assessment that must be done with YOUR OWN WORK! This Knowledge Check must be done with YOUR OWN WORK! (1) Find a basis of the kernel of T. (solution) Page 2 of 3 (K) Find the range of T. (solution) WINTER 2023 (J) Find the coordinates of 3t² - 4t relative to the basis you found in (1) above if applicable. If not, explain why not. (solution) (J) Find the coordinates of 3t² - 4t relative to the basis you found in (I) above if applicable. If not, explain why not. (solution) (K) Find the range of T. (solution) (L) Find a basis of the range of T. (solution) (M) Find the dimension of the range of T. (solution) of [77]ift (N) Find all the preimage(s) of (solution) if there is any. (O) Determine whether T' is on-to-one or not. Explain briefly. (solution) (P) Determine whether T is onto or not. Explain briefly. (solution) 1. (17 points) Consider the following set: S = = { ["@PO], where p(t) = at² + bt + c. [p(0) p(0)] (A) Show that the set is non-empty. (solution) (B) Determine whether there exists the additive identity of the set or not. If yes, find it. Otherwise, explain why not. (solution) (C) Determine whether the set is closed under addition or not. Explain why or why not. (solution) (D) Determine whether the set is closed under scalar multiplication or not. Explain why or why not. (solution) (E) Determine whether the set is a vector space or not. Explain why or why not. (solution) (F)-(P) Let T: P2 → S be defined by T (p(t)) = PO PO, where P₂ = {at² + bt + c, where a, b and c are any real numbers.}, and S is the set defined above. (G) Determine whether T is linear or not. Explain why or why not. (solution) (H) Find the kernel of T. (solution) This is an assessment that must be done with YOUR OWN WORK! This Knowledge Check must be done with YOUR OWN WORK! (1) Find a basis of the kernel of T. (solution) Page 2 of 3 (K) Find the range of T. (solution) WINTER 2023 (J) Find the coordinates of 3t² - 4t relative to the basis you found in (1) above if applicable. If not, explain why not. (solution) (J) Find the coordinates of 3t² - 4t relative to the basis you found in (I) above if applicable. If not, explain why not. (solution) (K) Find the range of T. (solution) (L) Find a basis of the range of T. (solution) (M) Find the dimension of the range of T. (solution) of [77]ift (N) Find all the preimage(s) of (solution) if there is any. (O) Determine whether T' is on-to-one or not. Explain briefly. (solution) (P) Determine whether T is onto or not. Explain briefly. (solution)

Answer rating: 100% (QA)

A To show that the set is nonempty we need to find at least one element that belongs to the set S Lets choose pt 2t2 3t 1 We can see that p0 1 so pt 1 1 2t2 3t 1 Therefore pt 2t2 3t 1 belongs to the s... View the full answer