Define the linear transformation T : P3 → P3, where P3 is the vector space of real values polynomials of degree at most

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Define the linear transformation T : P3 → P3, where P3 is the
vector space of real values
polynomials of degree at most 3, by the following formula.
p(x) 7→xp′′(x) −5p′(x) + 3p(x)
Here p′′(x) is the second derivative of p.
(a) Compute the matrix M(T) with respect to the basis {1,x,x2,x3}.
You don’t need to
explain how you did this computation, just give the matrix.
(b) Suppose that φ ∈ P3′
is defined by φ(p) = p′(4). Describe φ as a linear
combination
of the of the basis vectors φ1,...φ4, where φi takes the ith basis
vector of P3 to 1,
and maps all the other basis vectors of P3 to zero. From this
linear combination you
should be 4 coordinates, such that when you put them in a row
vector [c1,c2,c3,c4]
And you multiply this row vector by the column vector [a,b,c,d],
you get φ(p) where
p(x) = a + bx + cx2 + dx3.
(c) Let T′ be the dual map of T. Describe the linear functional
T′(φ) ∈P3′
. This should
be a linear map P3 → Rso your description should include what T′(φ)
does to a an
arbitrary polynomial q(x).
(d) Multiply [c1 ...c4]M(T). How is your answer related to what you
computed for T′(φ)