Math 4355, Spring 2022

Methods of Applied Mathematics

John Zweck

For starred problems, see hints at the bottom of this page.
L4H4.7A means go to Lecture 4 and find the problem label Hwk 4.7A.
L12CScI means go to Section C of Lecture 12 and do the Scaffolded Problem I
When you solve hwk problems, you should only use results from the section of the book the problem comes from, or from previous sections. Do not use results we have not yet discussed that you recall from your Linear Algebra course.

**Schedule**
Date	Section [of Meyer unless otherwise indicated]	Topic	Homework	Homework Due Date
-	1.2, 1.3, 2.1-2.5, 3.2	Review for self (grey boxes)	1.2.1, 1.2.8, 1.2.9, 1.3.1, 2.1.1ab, 2.2.1.a, 2.5.1ab, 3.2.3	Th Jan 27
W Jan 19	Day One 4.1	Subspaces	4.1: 1cde, 2acfh, 6de, 9, 11*	Th Jan 27
M Jan 24	3.3-3.6	Linear Transformations Matrix Algebra	3.2.6, 3.3.1adf 3.5.2, 3.5.3, 3.5.4, 3.5.7, 3.6.1, 3.6.2, 3.6.5*	Th Feb 3
W Jan 26	2.1, 2.2 3.9	Row Echelon Forms, Rank Elementary Matrices	2.1.5, 2.4.4, 2.4.5	Th Feb 3
M Jan 31	2.4 4.2	Homogeneous Systems Nullspace	-	Th Feb 10
W Feb 2	2.5 3.7 3.9 4.2	Nonhomogeneous Systems, Range Matrix Inversion Rank Normal Form Four Fundamental Subspaces	3.7: 3,8 3.9: 1, 2, 4, 6* 4.2: 1,2,3,7,10,12	Th Feb 10
M Feb 7	4.3	Linear Independence	2,6,7,10,12 Matlab Hwk #A (Due Feb 27)	Th Feb 17
W Feb 9	4.4	Basis, Dimension	1,4,5,8,9	Th Feb 17
M Feb 14	- 4.7	Matlab Demo Linear Transformations	-	Th Feb 24
W Feb 16	4.7	Linear Transformations	L4H4.7A, 6, 8, 9, 10, 12b,13,19	Th Feb 24
M Feb 21	4.8	Change of Basis Similarity	1,3,5,8,12	Th Mar 3
W Feb 23	5.3 5.4	Inner Product Spaces Orthogonal Vectors	5.3: 1*, 2, 4 5.4: 3, 4, 12, 14, 15	Th Mar 10
M Feb 28	5.5	Gram-Schmidt	1ab, 2, 3, 5, 11	Th Mar 10
W Mar 2	5.6	Unitary and Orthogonal Matrices	-	-
F-Su Mar 4-6	EXAM 1	-	-	-
M Mar 7	5.6	Unitary and Orthogonal Matrices	L8H5.6AA, L8H5.6A, L8H5.6B, L8H5.6C, 1, 2, 3, 5, 13, 18 Matlab Hwk #B (Due Apr 6)	M Mar 28
W Mar 9	5.9	Complementary Subspaces	L9H5.9A 1, 4, 5, 7, 11	M Mar 28
SPRING BREAK	-	-	-	-
M Mar 21	5.11	Orthogonal Decomposition	1,3,4,5,11	Th Apr 7
W Mar 23	7.1	Eigensystems	L12CSc, L12ESc1, L12ESc2, L12ESc3, 1bd, 8, 9, 10, 15	Th Apr 7
M Mar 28	7.2	Diagonalization	L13BScI, L13BScII, L13BScIII L13ESc 3, 5, 6	W Apr 13
W Mar 30	7.5	Spectral Theorem for Normal Matrices	-	-
M Apr 4	7.5	Spectral Theorem for Normal Matrices	L14ScB, L14ScC 1,3,13a	W Apr 13
W Apr 6	7.3 7.4	Functions of Diagonalizable Matrices Matrix Exp and ODEs	L16ExA, L16ExB 7.3: 3, 6 7.4: 1 Matlab Hwk #C (Due Apr 29)	Th Apr 21
M Apr 11	-	Fourier Series I	-	-
W Apr 13	-	Fourier Series I	FS Hwk #1 [pdf]	Th Apr 21
F-Su Apr 15-17	EXAM 2	-	-	-
M Apr 18	-	Fourier Series IIA	-	-
W Apr 20	-	Fourier Series IIB	FS Hwk #2 [pdf] Matlab Hwk #D (Due May 5, Extra Credit)	Th Apr 28
M Apr 25	-	Fourier Series III	-	Th May 5
W Apr 27	-	Discrete Fourier Transform	FS Hwk #3 [pdf]	Th May 5
M May 2	-	Filtering and Convolution	-	-
W May 4	-	Fourier Series IV	-	-
W May 11-F May 13	FINAL	-	-	-

Homework Hints

4.1.1: For this and other problems in which you have to identify whether a statement is true or not, either prove the statement is true or provide a simple counterexample with accompanying explanation.
4.1.11: To show two sets are equal show every element of one set is contained in the other, and vice-versa. Convert statements about sets into equations inviolving vectors in those sets.
3.2.6b: This is analogous to the proof that any function can be uniquely expressed as the sum of an odd and an even function. Hint for uniqueness: Show that if a square matrix is both symmetric and skew-symmetric then it is zero. You can do this just using matrix algebra, that is working with the matrix as an entity on its own right. It is not necessary to manipulate the entries of the matrix.
4.1.6de: See Example 4.1.7 Meyer pages 165-166.
3.5.3: Use the grey box on Meyer page 98.
3.6.5: See hint to 4.1.1
3.9.6: See hint to 4.1.1
4.2.2b: Here is a rephrasing of the question in case you need it. Suppose that Ax=b is consistent. Prove that Ax=b has a unique solution if and only if N(A) = {0}.
4.2.7: Show that if Ax=0 then the length of x is zero.
4.3.6: First understand why result is true in 3x3 case. Then prove in general.
4.3.10: Find a vector in the nullspace of A
5.3.1: See hint to 4.1.1