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CSE 313 Final Examination - Mathematics Questions, Exams of Linear Algebra

Chitkara UniversityLinear Algebra

The final examination questions for a cse 313 course, focusing on mathematics. The questions cover topics such as vector balancing, differential equations, matrix determinants, and coordinate transformations. Students are required to derive expressions, determine stability, prove mathematical statements, and solve for coordinates. The document may be useful for university students preparing for exams or quizzes, particularly those in the field of computer science or engineering.

Typology: Exams

2012/2013

Uploaded on 02/14/2013

ashwini
ashwini 🇮🇳

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CSE 313
Final Examination
May 7, 2004
Question 1: {20 pts}
Derive an expression for the scalar,
γ
, as a function of the vectors,
n
Rvu ∈,
, that will balance the following equation. (Make sure to
show your work)
))(()( 1TT uvIuvI γ−=− −
Use this result to invert the following matrix:
−
−−
−−
0011
2122
0010
1012
Question 2: {15 pts}
Determine whether the solutions to the following differential
equation are stable or unstable:
xxx 45 −−=
Question 3: {15 pts}
If A is a square matrix prove that the absolute value of the
determinant of A,
)det( A
, is equal to the product of its singular
values.
pf3

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CSE 313

Final Examination

May 7, 2004

Question 1: {20 pts}

Derive an expression for the scalar, n γ^ , as a function of the vectors,

u , v∈ R , that will balance the following equation. (Make sure to

show your work)

( I − uvT^ )−^1 =(I−γ(uvT ))

Use this result to invert the following matrix:

Question 2: {15 pts}

Determine whether the solutions to the following differential

equation are stable or unstable: x^ =−^5 x−^4 x

Question 3: {15 pts}

If A is a square matrix prove that the absolute value of the

determinant of A, det(^ A)^ , is equal to the product of its singular

values.

Question 4: {10 pts}

If y^ ∈^ Range(^ A) and z^ ∈^ Null(^ AT)for some matrix, A ∈ Rm×n, show that

yT^ z= 0.

Question 5: {15 pts}

If {^ v 1 vn} are orthonormal vectors, v^ i ⋅^ vj= 01 ififii=≠jj, and x^ =^ ∑i αi^ vi

show that x^2 =^ ∑i α^ i^2.

Based on this result, explain why dropping small Fourier

coefficients is an effective strategy for compressing audio signals.

Question 6: {15 pts}

Consider the figure shown below, let x^ A ,^ yA denote the coordinates of

a point with respect to frame A and xB^ ,^ yB denote the coordinates of

the same point with respect to frame B. These coordinate values can

be related by the following equation. 

1 1 B

B A AB yxA g yx

θ

1 Β Α

Give an expression for the 3 by 3 matrix g^ AB.

Let g^ AT denote the 3 by 3 matrix that relates coordinate frames A

and T in the figure below. Using your previous result, express g^ AT as