MATH quiz (need it in 8 hours)

1: (10 Points) Prove or disprove the following statements (if it is true, please write a proof; if it is false,

please provide an counter example to disprove it):

a:

Let S = {t2 + 1, t + 1, t2 + t} be a set of polynomials. Then P2 = Span(S).

b:

The set of all pairs of real numbers of the form hx, yi, where xy � 0, with the

standard operations on R2 is a vector space. (That is the union of the first and third

quadrants in the xy-plane.)

2

2: (15 Points)Let T : R7 ! R4 be given by T(~x) = A~x,

where A =

2

66 4

7 �28 2 17 �3 73 24 �3 �12 4 �17 2 �3 �22 �5 20 2 �19 7 �67 �84 2 �8 �3 12 4 �43 37

3

77 5

rref���!

2

66 4

1 �4 0 3 0 5 6 0 0 1 �2 0 7 �3 0 0 0 0 1 �8 4 0 0 0 0 0 0 0

3

77 5

.

a: Determine row space of A as span of the linearly independent vectors.

b: Determine column space as span of the linearly independent vectors.

c: Determine ker(T) or the null space as span of the linearly independent vectors.

3

3: (45 Points) Consider the matrix A =

2

4 7 4 16

�2 �2 �4 �4 �2 �9

3

5

a: (12 Points) Find the characteristic polynomial. Then find the eigenvalues by solving

the characteristic polynomial.

4

b: (12 Points) Find a basis for each eigenspace corresponding to each eigenvalue from

part a. Indicate the dimension of each eigenspace for the matrix A.

5

c: (4 Points) Diagonalize the matrix A, that is, find an invertiable matrix P , and a

diagonal matrix D such that A = PDP �1.

d: (10 Points) Find the inverse of P by performing row operations to the matrix [P |I].

Then perform matrix multiplication to verify that A = PDP �1.

6

e: (7 Points) Use the factorization A = PDP �1 to prove that A7 = PD7P �1. Then

compute A7. Make sure you compute the following separately: D7, PD7.

7

4: (10 Points) The first four Chebyshev Polynomials of the Second kind are 1, 2t, �1 + 4t2, and �4t +

8t3.

a: Show that the first four Hermite polynomials form a basis of P3.

b: Denote the Chevyshev Polynomials as B and let p(t) = 1 + 8t2 + 8t3. Find the

coordinate vector of p(t) relative to B, where B is the basis of P3 consisting of the

Chebyshev Polynomials.

8

5: (20 Points) Consider the matrix A =

2

66 4

�1 6 6 3 �8 3 1 �2 6 1 �4 �3

3

77 5.

a: Show that the column vectors are linearly independent; and hence, we can conclude

that the column space is of dimension 3.

b: By the Gram-Schmidt Process, find an orthogonal basis for the column space of the

matrix A

9