The function T : 1R2 } R3 is dened by Show that T is linear. To malte sure you are on the right track you should answer the following questions. a. Is the domain of T a known vector space? I Yes 1" I is. Is the codomain of T a known vector space? I Yes .- I c. Does T preserve vector addition? I Yes 1" I d. Does T preserve scalar multiplication? I Yes 1" I Show that S = XER4 : 821 +812 +513 + 314 =0. is a subspace of IR4. To check that you are on the right track, answer the following questions. . Is S a subset of a known vector space? Yes . Does S contain the zero element? Yes . Is S closed under vector addition? Yes . Is S closed under scalar multiplication? YesLet In be the set of real polynomials of degree at most n. Show that S= PEPA : 12 + x - 1 is a factor of p(I) is a subspace of P4 To check that you are on the right track, answer the following questions. . Is S a subset of a known vector space? Yes . Does S contain the zero element? Yes . Is S closed under vector addition? Yes . Is S closed under scalar multiplication? YesThe air in a 41 cubic metre kitchen is initially clean, but when Thanom burns her toast while making breakfast, smoke is mixed with the room's air at a rate of 0.06 mg per second. An air conditioning system exchanges the mixture of air and smoke with clean air at a rate of 4 cubic metres per minute. Assume that the pollutant is mixed uniformly throughout the room and that burnt toast is taken outside after 32 seconds. Let S(t) be the amount of smoke in mg in the room at time t (in seconds) after the toast first began to burn. a. Find a differential equation obeyed by S(t). b. Find S(t) for 0