Questions in Linear algebra

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please read the questions carefully and give a detailed answer.

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(1) Let T:R2 + R2 be the linear map given in standard coordinates by the matrix
Let B = {(1,0),(0,1)} and B’ = {(-2, 1), (2,1)}.
(a) Find the change of basis matrices PBB’ and PB’s and use these to compute the matrix of T
relative to B’ i.e. the above matrix is TBB, and use PBB’ and Pb’s to find TB’B).
(b) Use TB or to find the kernel and image of T.
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(2) Let T: P2 → R2 be given by
T(p) =
(e.g. if p= a +bx, then p(4) = a +b(4) = a + 46.)
(a) Find the matrix of T relative to the standard bases B = {1, 2, 22} of P2, and C = {el, ez} of
R2.
(b) Find the matrix of T relative to the basis A = {1,1+x,1++} of P2 and D= {(1, 1), (1, -1)}
of R2
(c) Find a basis for Kert and Im T using any method you wish.
(3) Suppose that T: P2 → P2 satisfies
T(1+x) = 3(x+x²), 1(x + x2) = -(+1), 7(2? + 1) = 2(x + x2)+(x + x2).
Caleulate KerT and Im T.
Find the matrix of T relative to the basis {1+, x + x2, 22 +1}, ealeulate kernel and image relative
to that basis, then rewrite the kernel and image as polynomials.)
(4) Suppose T : R2 + R2 satisfies
T(3, 1) = 5(3,1), T(0,2) = -1(0,2).
Find the matrix of T relative to the standard basis of R2.
(5) Suppose that a matrix A, B e Mnxn is diagonal; i.e. the entry in the i-th row, j-th column
i=j
lbi i=j
ij
it i
di
Ajj =
Bij =
Prove that AB is diagonal with
(AB)ij =
ſazbi i=;
lo itj