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This is a final exam for a linear algebra 2 course, consisting of 10 questions including concept problems, finding linear transformations, matrix computation, inner product space problems, and gram-schmidt process. The exam also includes finding the adjoint of a linear transformation and proving the triangle inequality.
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1/10 (Each 5 points) Concept Problems.
(1) Write down the de nition of a linear transformation on a vector space.
(2) Write down the necessary and sucient condition(s) for a linear transformation T : V! V of the nite dimensional vector space V to be diagonalizable.
(3) Write down the Riesz Representation Theorem.
(4) Write down the Spectral Theorem.
4/10 (5 points) Let T : P 2! R^3 be de ned by T (a + bx + cx^2 ) = (2a + c; a b c; 3 b + c). Find the matrix [ T ]EE 0 of T relative to E and E^0. Here E =
e 1 = 1; e 2 = x; e 3 = x^2 and E^0 = fe^01 = (1; 0 ; 0); e^02 = (0; 1 ; 0); e^03 = (0; 0 ; 1)g are the standard bases of P 2 and R^3 , respectively.
5/10 (Each 5 points) Let T : R^3! R^3 be a linear transformation de ned by
T (x; y; z) =
x y p 2
y x p 2
; z
(1) Compute T 2 , i.e., nd T 2 (x; y; z). Here T 2 means T 2 (x; y; z) = T (T (x; y; z)).
(2) Find the characteristic polynomial and eigenvalues and eigenspaces of T 2.
(3) Diagonalize T 2 , if possible.
(4) Compute T 2 n, i.e., nd T 2 n(x; y; z) for a positive integer n.
(5) When considering R^3 as the standard inner product space, nd the orthonormal basis for T 2 , if possible.
8/10 (5 points) Let R^3 be the standard inner product space. Given the set f e 1 = (1; 1 ; 0); e 2 = (0; 2 ; 1); e 3 = (3; 1 ; 4) g R^3 , deduce an orthogonal set by using the Gram{Schmidt process.
9/10 (5 points) Let T : R^3! R^3 be a linear transformation de ned by T (x; y; z) = (2x; 3 x; 4 y 3 z). When considering R^3 as the standard inner product space, nd the adjoint T ^ of T , i.e., nd T (x; y; z). Is T self{adjoint? Justify your answer.
10/10 (10 points) Let P 2 be the inner product space where P 2 is the set of polynomials of degree at most 2 and the inner product is de ned by h p(t); q(t) i =
1
p(t)q(t) dt:
Let " : P 2! R be a linear transformation de ned by "(p(t)) = p(1) p( 1). Find q(t) 2 P 2 such that for all p(t) 2 P 2 ,
p(1) p( 1) = "(p(t)) = h p(t); q(t) i =
1
p(t)q(t) dt; i:e:; p(1) p( 1) =
1
p(t)q(t) dt:
