# HOMEWORK 01 Solution

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## Description

• Read Chapters 1 & 2 in Eldén.

• Read Lecture Notes 01–04

• Given the prerequisites for this class – a strong knowledge of Linear Algebra – and the material outlined in the course syllabus, the following concepts are important for you to understand. We will discuss some of these concepts in my lectures, but do not wait for these lectures. If your memory is vague on these subjects, review them yourself now!

Solving linear systems

LU decomposition

Equivalences of non-singular matrices

Rank and span

Domain & Codomain

Domain, range

Column space and row space

Nullspace (Kernel) and Left Nullspace,

Linear combination

Basis, Change of basis

Orthogonal vectors, Gram-Schmidt orthogonalization

Inner product (dot product) and outer product

Eigenvalues, eigenvectors, diagonalization, eigenbasis

Problem 01 This is a simple MATLAB exercise.

(a) Download the data file: HW_01.mat to your working directory, and load it into your MATLAB session by:

>> load HW_01;

Then, draw the signal x in the data file using the following commands:

• figure(1);

• stem(x); hold on; plot(x); grid;

Note that this signal x consists of only 8 points, i.e., a very short signal (vector).

(b) In a different figure window, draw the 8 basis vectors stored as column vectors of the matrix U as follows:

© Professor E. G. Puckett – 1 – Revision 2.00 Wed 11th Apr, 2018 at 14:02

MAT 167–001 HOMEWORK 01 SPRING QUARTER 2018

• figure(2);

• for k=1:8 subplot(8,1,k);

stem(U(:,k)); axis([0 9 -0.5 0.5]); axis off; hold on;

end

• for k=1:8 subplot(8,1,k); plot(U(:,k));

end

You may need to see the details of these 8 plots by enlarging the window to a full screen. Print this figure and attach it to your HW submission.

(c) Compute the expansion coefficients (i.e., the weights of the linear combination) of x with respect to the basis vectors U(:, 1), . . . , U(:, 8) via

>> a=U’*x;

(d) Check the values of the entries of the coefficient vector a and create a new vector a2 of length 8 whose only nonzero entries are the two largest entries of a in terms of their absolute values.

(e) Construct an approximation x2 of x using a2. Then, plot x2 over Figure 1 as follows:

>> figure(1); stem(x2,’r*’); plot(x2,’r’);

(f) Now, instead of a2, let’s construct a4 of length 8 whose only nonzero entries are the four largest entries of a in terms of their absolute values. Then,

(g) Construct an approximation x4 of x using a4. Then, plot x4 over Figure 1 as follows (note using the different color from x2):

>> figure(1); stem(x4,’gx’); plot(x4,’g’);

Then, print out Figure 1, and attach it to your HW submission.

(h) Consider now x8, which is just a full reconstruction without throwing out any coefficients, i.e.,

>> x8=U*a;

Finally, compute the relative error of x8 by

>> sqrt(sum((x-x8).^2)/sum(x.^2))

and report the result. Similarly compute the relative error of x4 and x2, and report the results.

(i) Write a detailed explanation of what this MATLAB program does.

© Professor E. G. Puckett – 2 – Revision 2.00 Wed 11th Apr, 2018 at 14:02

MAT 167–001 HOMEWORK 01 SPRING QUARTER 2018

Problem 02 Consider the following set of terms (words) and documents (or rather book titles):

 Terms Documents T1: Book (Handbook, BOOK) D1: The Princeton Companion to Mathematics T2: Equation (Equations) D2: NIST Handbook of Mathematical Functions T3: Function (Functions) D3: Table of Integrals, Series, and Products T4: Integral (Integrals) D4: Linear Integral Equations T5: Linear D5: Proofs from THE BOOK T6: Mathematics (Mathematical) D6: The Book of Numbers T7: Number (Numbers) D7: Number Theory in Science and Communication T8: Series D8: Green’s Functions and Boundary Value Problems D9: Discourse on Fourier Series D10: Basic Linear Partial Differential Equations D11: Mathematical Physics, An Advanced Course

(a) Construct 8 £11 term-document matrix.

(b) Suppose we want to query “Integral Equation.” Construct the query vector q.

(c) Find three closest documents for the query in (b).

Problem 03 (This is a review problem.) At the beginning of 2009, the population of California was 36,453,973. The pop-ulation living in the United States but outside of California was 271,491,582. During that year, 458,682 people moved to California from another state. Similarly, 545,921 people moved from California to elsewhere in the United States. Set up a matrix vector multiplication problem whose solution shows the population changes in California and in the rest of the United States for 2009.

Problem 04 (This is another review problem.) Use the Gram-Schmidt Process to construct an orthonormal basis for R4 starting with the following set of vectors.

(a)

 203 2 1 3 203 213 6 1 7 6 0 7 6 1 7 6 1 7 x1 ˘ 1 , x2 ˘ 0 , x3 ˘ 0 , x4 ˘ 1 6 0 7 6 1 7 6 1 7 6 1 7 6 7 6 ¡ 7 6 7 6 7 4 5 4 5 4 5 4 5

(b)

 203 213 203 223 6 1 7 4 7 6 1 7 6 0 7 y1 ˘ 0 , y2 ˘ 60 , y3 ˘ 2 , y4 ˘ 0 6 1 7 60 7 6 1 7 6 1 7 6 7 6 7 6 7 6 7 4 5 4 5 4 5 4 5

© Professor E. G. Puckett – 3 – Revision 2.00 Wed 11th Apr, 2018 at 14:02

MAT 167–001 HOMEWORK 01 SPRING QUARTER 2018

Problem 05 (This is also a review problem.) Consider the following matrix:

212343

601577

• 7

600167

• 7

660 0 0 177

• 7

600007

• 7

 4 0 0 0 0 5 0 0 0 0

(a) View this matrix as a linear transformation, T, between two vector spaces. What are the domain and the codomain of T?

(b) Identify (i.e., define) the column space, C(A), (or image) of this linear transformation.

(c) Find a basis for the column space, C(A), of T.

(d) Identify the nullspace, N(A), (also known as the kernel) of T.

(e) Write a basis for the nullspace, N(A), of T.

(f) What is the row space of this linear transformation?

(g) What is the left null space N(AT), of T.

(h) What is the rank of the matrix A?

Problem 06 Convert the following numbers to from base X to base Y as specified below.

(a) Convert the following binary numbers to decimal:

1. 11111

1. 1000000

1. 1001101101

1. 10101010

1. 000011110000

(b) Convert the following decimal numbers to binary:

1. 73

1. 127

1. 402

1. 512

1. 1000

1. 32767

© Professor E. G. Puckett – 4 – Revision 2.00 Wed 11th Apr, 2018 at 14:02

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