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CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
18-771
_
LINEAR SYSTEMS
SPRING 2002
_
MIDTERM EXAMINATION
OPEN TEXTBOOK AND COURSE NOTES
TUESDAY 26 FEBRUARY 2002
2:30 P.M. – 4:20 P.M.
PH A22
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OVERVIEW
This midterm examination (which constitutes thirty-five percent of your final grade)
consists of five (5) problems and will be graded on the basis of three hundred fifty (350)
points. You are to solve these problems in the blue books. Present your solutions in
consecutive order and put a box around your answers.
The objective of this midterm examination is to test your understanding of statespace fundamentals and your ability to apply these concepts to solve engineering
problems. Most of the tasks do not require extensive computation. Wherever appropriate,
solutions based upon engineering insight will receive full credit.
The quality of your analysis, synthesis and evaluation is as important as your
answers. Your reasoning must be precise and clear; your complete English sentences
should convey what you are doing. Your grade will reflect both the clarity of your
presentation and checking of your results.
A grading sheet appears on page 6. Letter your name on the cover of each blue
book and the grading sheet. Submit the grading sheet along with your blue books.
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PROLOGUE
For each problem, formulate a coherent plan to gain insight, generate
straightforward algebraic expressions and extract the salient information required to
accomplish the prescribed tasks. To alleviate the computational tedium, carry your
algebraic computations to the point of addressing the issues and drawing engineering
conclusions. To facilitate your algebraic computations, you are encouraged to define
intermediate parameters and variables; e.g., g  g1 + g2. You are not required to simplify
algebraic expressions for the aesthetic objective of simplifying algebraic expressions.
PROBLEM I (90 POINTS)
Do all of the Parts of this Problem Without a Calculator.
In the system of linear algebraic equations
Ax = b
(1)
the coefficient matrix
 1 3 0 2  1


A  0 0 1 4  3
 1 3 1 6  4


and the right-hand side input vector
 b1 
 
b   b2 
b 
 3
(A – 10 Points)
Find the rank of A.
(B – 20 Points)
List the pivot columns of A.
(C – 10 Points)
State the solvability condition for the system of linear algebraic
equations in (1).
(D – 10 Points)
Find a basis for the null space of A.
(E – 10 Points)
Find a basis for the column (range) space of A.
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(F – 10 Points)
Find the most general solution of the system of linear algebraic
equations in (1)
(G – 10 Points)
Exhibit the structure of the singular value matrix ∑ in the economy size
singular value decomposition of the coefficient matrix A in (1).
(H – 10 Points)
Write MATLAB code to apply Gaussian elimination to compute
numerically the solution of the system of linear algebraic equations in
(1).
PROBLEM II (50 POINTS)
An engineer is attempting to orthogonalize (orthonormalize) the columns of the
( 2 x 2 ) matrix
 0.70000 0.70711
A  

 0.70001 0.70711
(1)
The engineer begins by implementing in floating point arithmetic the classical and
modified Gram – Schmidt (G – S) algorithms on a computer that rounds all computational
results to five digits of relative accuracy. For ( 2 x 2 ) matrices, the classical and modified
Gram – Schmidt algorithms are identical. The engineer computes
 0.70710 1.0000 

Q  
 0.70711 0.0000 
(2)
The engineer proceeds to test the following MATLAB – script on a computer with
sixteen-digit precision. eye(2) is the ( 2 x 2 ) identity matrix.
A = [.70000 .70711
.70001 .70711];
[Q,R] = qr(A);
norm(Q’*Q-eye(2))
[Q,R] = mgs(A);
norm(Q’*Q-eye(2))
%Define A.
%Compute Q by Householder.
%Test orthogonality of Q.
%Compute Q by modified G-S.
%Test orthogonality of Q.
The lines without semicolons produce the following printed output:
ans = 2.3515e-16
ans = 2.3014e-11
The engineer hires you to accomplish the following tasks and assess the quality of
the computational results.
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(A – 25 Points)
Evaluate quantitatively the orthogonality of the Q matrix in (2). Justify
your evaluation.
(B – 25 Points)
Interpret the MATLAB output and draw your conclusions.
PROBLEM III (55 POINTS)
Let
 1 0 0

1 1 0
M(4)  
1 1 1

1 1 1
1

1
1

1
(A – 30 Points)
Compute manually the LU decomposition of M(4). Show all steps.
(B – 15 Points)
Generalize and comment on your findings for the ( n x n ) matrix M(n).
(C – 10 Points)
Write MATLAB code to compute the LU decomposition of M(n).
PROBLEM IV (55 POINTS)
Let A be an ( n x n ) symmetric positive definite matrix. Write computationally efficient
MATLAB – script to evaluate the quadratic form
Q (x) = xT A-1 x
Comment on the efficacy of your algorithm.
PROBLEM V (100 POINTS)
Matrix factorizations and decompositions abound throughout numerical linear
algebra. Familiar examples include the LU factorization, Cholesky decomposition, QR
factorization and singular value decomposition of the matrix A. The objective of this
problem is to formulate and illustrate an algorithm for computing the full rank factorization
of the rank r matrix A [ m x n ] .
The goal is to compute the rank r matrices F [ m x r ] and G [ r x n ] such that
A = FG
(1)
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The factorization in (1) is called the full rank factorization of A. Intuition correctly tells
us that all [ m x n ] matrices A with rank r > 0 have a full rank factorization.
(A – 50 Points)
Formulate a MATLAB – oriented algorithm for computing the full rank
factorization of the rank r matrix A [ m x n ] .
(B – 50 Points)
Apply your algorithm to compute manually the full rank factorization of
 3 6 13 


A [ 3 x 3 ]  2 4 9 
1 2 3 


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CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
18-771
_
LINEAR SYSTEMS
SPRING 2002
_
MIDTERM EXAMINATION
GRADING SHEET
TUESDAY 26 FEBRUARY 2002
NAME
SCORE
PROBLEM I (90 POINTS)
PROBLEM III (55 POINTS)
(A – 10 POINTS)
(A – 30 POINTS)
(B – 20 POINTS)
(B – 15 POINTS)
(C – 10 POINTS)
(C – 10 POINTS)
(D – 10 POINTS)
(E – 10 POINTS)
PROBLEM IV (55 POINTS)
(F – 10 POINTS)
(G – 10 POINTS)
PROBLEM V (100 POINTS)
(H – 10 POINTS)
(A – 50 POINTS)
(B – 50 POINTS)
PROBLEM II (50 POINTS)
(A – 25 POINTS)
(B – 25 POINTS)