Complex Inner Product - Matrix Methods - Exam | Exams Mathematics

APPM 3310: Matrix Methods — Exam #2 — November 12, 2008

On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading

table. Explain all of your answers. A correct answer with no supporting work may receive no credit

while an incorrect answer with some correct work may receive partial credit. Start each problem

on a new page. No books, notes or electronic devices of any kind (e.g. cell phones, calculators, etc.)

are permitted. SHOW ALL WORK.

1. (20 points)

(a) Suppose Vis a complex vector space and that hv,wiis a pairing that takes two vectors

v,w∈Vand produces a complex number hv,wi ∈ C. What three conditions must hv,wi

satisfy to be considered a complex inner product on the vector space V? (Explicitly state

your answer, don’t just name the properties.)

(b) Does hv,wi=v1w1+ 2v2w2define a complex inner product on C2? (Show all work.)

conditions must kvksatisfy to be considered a norm on the vector space V?(Explicitly state

your answer, don’t just name the properties.)

2. (20 points)

(a) State the Cauchy-Schwarz Inequality. When does equality hold?

(b) State the Triangle Inequality. When does equality hold?

to the L2inner product on [−1,1].

3. (20 points)

(a) Let Vbe an inner product space. State what it means for an n×nmatrix Kto be the

Gramm matrix associated to v1, . . . , vn∈V.

(b) Under what conditions will the Gramm matrix Kbe positive definite? Under what condi-

tions will Kbe positive semi-definite?

4. (20 points)

(a) Prove that a positive definite n×nmatrix Khas positive determinant: detK > 0.

(b) Prove that every 2 ×2 symmetric matrix with positive determinant and postive trace is

positive definite.

zfor z∈Cforms a subspace

of C2. (Justify your answer.)

(d) True or False: Reordering the original basis before starting the Gram-Schmidt process

leads to the same orthogonal basis. (Justify your answer.)

5. (20 points) Find orthonormal bases for the four fundamental subspaces associated with the

matrix A=



1 0 1 0

1 1 1 1

−1 2 0 1



.

6. (10 points) Find the closest vector, v?, on the subspace spanned by (0,0,1,1)Tand (2,1,1,1)T

to the vector b= (0,3,1,2)T.

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APPM 3310: Matrix Methods — Exam #2 — November 12, 2008 On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading table. Explain all of your answers. A correct answer with no supporting work may receive no credit while an incorrect answer with some correct work may receive partial credit. Start each problem on a new page. No books, notes or electronic devices of any kind (e.g. cell phones, calculators, etc.) are permitted. SHOW ALL WORK.

(20 points) (a) Suppose V is a complex vector space and that 〈v, w〉 is a pairing that takes two vectors v, w ∈ V and produces a complex number 〈v, w〉 ∈ C. What three conditions must 〈v, w〉 satisfy to be considered a complex inner product on the vector space V? (Explicitly state your answer, don’t just name the properties.) (b) Does 〈v, w〉 = v 1 w 1 + 2v 2 w 2 define a complex inner product on C^2? (Show all work.) (c) Suppose V is a vector space and suppose for each v ∈ V we have ‖v‖ ∈ R. What three conditions must ‖v‖ satisfy to be considered a norm on the vector space V ?(Explicitly state your answer, don’t just name the properties.)
(20 points) (a) State the Cauchy-Schwarz Inequality. When does equality hold? (b) State the Triangle Inequality. When does equality hold? (c) Verify the Cauchy-Schwarz inequality for the functions f (x) = x and g(x) = ex^ with respect to the L^2 inner product on [− 1 , 1].
(20 points) (a) Let V be an inner product space. State what it means for an n × n matrix K to be the Gramm matrix associated to v 1 ,... , vn ∈ V. (b) Under what conditions will the Gramm matrix K be positive definite? Under what condi- tions will K be positive semi-definite? (c) Prove that every positive definite n × n matrix K can be written as a Gramm matrix.
(20 points) (a) Prove that a positive definite n × n matrix K has positive determinant: det K > 0. (b) Prove that every 2 × 2 symmetric matrix with positive determinant and postive trace is positive definite. (c) True or False: The set of complex vectors of the form

(z z

for z ∈ C forms a subspace of C^2. (Justify your answer.) (d) True or False: Reordering the original basis before starting the Gram-Schmidt process leads to the same orthogonal basis. (Justify your answer.)

(20 points) Find orthonormal bases for the four fundamental subspaces associated with the matrix A =

^11 01 11

(10 points) Find the closest vector, v?, on the subspace spanned by (0, 0 , 1 , 1)T^ and (2, 1 , 1 , 1)T to the vector b = (0, 3 , 1 , 2)T^.

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^11 01 11