# Q3. Consider the vector space P, consisting of all polynomials of degree at most two together...

###### Question:

Q3. Consider the vector space P, consisting of all polynomials of degree at most two together with the zero polynomial. Let S = {p.(t), p2(t)} be a set of polynomials in P, where: pi(t) = -4 +5, po(t) = -3° - 34+5 (a) Determine whether the set S = {P1(t).pz(t)} is linearly independent in Py? Provide a clear justification for your solution. (8 pts) (b) Determine whether the set S = {p(t),p2(t)} spans the vector space P ? Provide a clear justification for your solution. (7 pts) (e) Is it possible to find a polynomial pz(t) in P, so that {pı(t), p2(t), ps(t)} is a basis for the vector space Ps? (Justify your answer).(5 pts)
-32 Q4. Let L: R R be a transformation defined by L uj - 2u2 U-U (a) Show that is a linear transformation. (8 pts) (1) Find the standard matrix A of L, and find 2 ([ using the matrix A. (7 pts) (c) Do you think that any transformation T: R R is linear? (Justify your answer).(5 pts)

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