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[answered] MATH 416 FINAL EXAM NAME: INSTRUCTIONS: Unless otherwise in


Could you please help me to do Q5&Q6 in the attached file?


MATH 416

 

FINAL EXAM

 

NAME: UIN: INSTRUCTIONS: Unless otherwise indicated, you must show all of your work to receive full

 

credit. Please write your answers in the space provided. No calculators, electronics, notes or other

 

aids are to be used while taking this test. (1) (10 points)

 

(a) (5 points) Find the set of all solutions of the following system of linear equations

 

x1 + x2 + x3

 

= 3

 

x1 + 2x2 + x3 ? x4 = 2

 

3x1 + 2x2 + 3x3 ? 2x4 = 1. (b) (5 points) For which values of B is the following system inconsistent

 

x1 + Bx2

 

= 0

 

x1 + 2x2 ? x3 = 0

 

x2 + x3 = 2. 2 (2) (20 points)

 

(a) (5 points) Prove that ? = {1 + 2x, x, x2 ? 3} is a basis for P2 (R). (b) (5 points) If ? = {1, x, x2 } is the standard basis for P2 (R) and T : P2 (R) ? P2 (R) is

 

the map

 

T (f (x)) = f 00 (0) + f 0 (0)x + f (0)x2

 

compute [T ]?? . 3 (c) (5 points) Determine if the map T from part (b) is invertible. (d) (5 points) If U : P2 (R) ? P2 (R) is a linear map and [U ]?? = I3 , find U (1), U (x), and

 

U (x2 ) and (hence) the formula for U (a + bx + cx2 ). 4 (3) (15 points)

 

(a) (10 points) Do FIVE of the following.

 

(i) State the Caley-Hamilton Theorem.

 

(ii) State the Rank-Nullity (Dimension) Theorem.

 

(iii) For A = (aij ) ? Mm?n , and B = (bij ) ? Mn?p give the formula for the ij th entry

 

of AB.

 

(iv) For A = (aij ) ? Mn?n , give a formula for det(A).

 

(v) Define the Frobenious inner product on Mn?n (F ).

 

(vi) State the Cauchy-Schwarz Inequality.

 

(vii) Define the generalized eigenspace of an eigenvalue ? of T : V ? V . (b) (5 points) Do ONE of the following.

 

(i) For A = (aij ) ? Mm?n , B = (bij ) ? Mn?p and c ? R prove that (cA)B = c(AB).

 

(ii) For A = (aij ) ? Mn?n and c ? R, prove that det(cA) = cn det(A).

 

(iii) For the Frobenious inner product on Mn?n (C) prove that

 

hA, Ai ? 0

 

with equality only if A = 0. 5  

 

1 3

 

(4) (12 points) Given the matrix A =

 

, derive a formula for An for any n ? N. Your

 

3 1

 

final answer should be in the form of a single 2?2 matrix. When you are done please rewrite

 

your answer here: An = . 6 (5) (17 points) Consider the vector space P1 (R) equipped with the inner product

 

Z 1

 

f (t)g(t)dt.

 

hf (x), g(x)i =

 

0 (a) (8 points) Construct an orthonormal basis for P1 (R). (b) (7 points) Consider W = P1 (R) to be a subspace of V = C([0, 1]), the vector space of

 

real-valued continuous functions on [0, 1]. Equip V with inner product

 

Z 1

 

f (t)g(t)dt.

 

hf (x), g(x)i =

 

0 Find the element u of W which is closest to h(x) = sin(x) ? C([0, 1]). (c) (2 points) Find an element z of W ? . 7 (6) (12 points) Let (V, h , i) be a finite dimensional inner product space and let T : V ? V be

 

a linear map.

 

(a) (2 points) What condition defines the adjoint map T ? : V ? V ? (b) (4 points) What is the relationship between the matrix representatives of T and T ? ?

 

Take care to define (or explain) all the relevant terms in your expression. (c) (2 points) Define what it means for T to be a normal operator. (d) (4 points) Consider V = M3?1 with its standard inner product and the linear map

 

LA : M3?1 ? M3?1 defined by LA (x) = Ax for 1 1 0

 

A = ?1 1 0 .

 

0 0 1

 

Determine if LA is normal. 8 (7) (35 points) Consider the matrix 2 0

 

A= 0

 

0 1

 

0

 

2

 

1

 

0

 

3

 

1 ?1 0

 

0 .

 

0 3 (a) (8 points) Compute the characteristic polynomial of A and verify that A has two district

 

eigenvalues, each with algebraic multiplicity 2. (b) (7 points) Determine if A is diagonalizable. 9 (c) (10 points) Find bases for the generalized eigenspaces of both eigenvalues of A. (d) (10 points) Construct a Jordan basis ? for A and write down the corresponding Jordan

 

canonical form of A. 10 (8) (12 points) Circle the statements which are TRUE.

 

(a) If W1 and W2 are both subspaces of V then so is their union W1 ? W2 .

 

(b) Every A ? M2?2 (C) has at least one eigenvalue.

 

(c) Every A ? M2?2 (C) is diagonalizable.

 

(d) Every inner product space with (nonzero) finite dimension has an orthonormal basis.

 

(e) If V is finite dimensional then the trivial map T0 : V ? W defined by T0 (x) = 0W for

 

all x ? V , has rank = dim(V ).

 

(f) If V admits a basis consisting of eigenvectors of T ? L(V, V ), then T is diagonalizable. Bonus Problem: (6 points) Provide proofs of the true statements above and explicit

 

examples (or clear arguments) demonstrating the falseness of the false statements. 11

 


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