207HW — Homework for CSUP Math 207, Fall 2011

Colorado State University, Pueblo; Fall 2011
Math 207 — Matrix and Vector Algebra with Applications
Homework Assignments & Course Schedule

Here is a link back to the course syllabus/policy page.

In the following all sections and page numbers refer to the required course textbook, Linear Algebra, A Modern Introduction (2nd ed.), by David Poole.

This schedule is subject to change, but should be accurate at any moment for at least a week into the future. Please check regularly to keep an eye out for changes.

Writing Emphasis problem(s) are indicated below as "17_WE", for example.

M:

Read: To the Student, p. xxiii and §1.1

Content:

bureaucracy and introductions

what are vectors

"=" for vectors

components

row- and column-vectors

HW0: Send me e-mail (at jonathan.poritz@gmail.com) telling me:

Your name.

Your e-mail address. (Please give me one that you actually check fairly frequently, since I may use it to contact you during the term.)

Your year/program/major at CSUP.

The reason you are taking this course.

What you intend to do after CSUP, in so far as you have an idea.

Past math classes you've had.

Other math and science classes you are taking this term, and others you intend to take in coming terms.

Your favorite mathematical subject.

Your favorite mathematical result/theorem/technique/example/problem.

Anything else you think I should know (disabilities, employment or other things that take a lot of time, etc.)

[Optional:] The best book you have read recently.

I will only enter your name into my gradebook when I get this e-mail, so you really need to do this assignment ASAP. Please take a moment to be complete (indeed, be as expansive as you can) — the more information I have about you, the better I can adapt the course to your needs and interests.
W:
- Read: §§1.1 & 1.2
- Content:
  - scalar multiplication
  - vector addition
  - properties of these vector operations
  - the norm of a vector
  - the dot product of two vectors
  - R², R³, and Rⁿ
  - a linear combination of two vectors
  - unit vectors
F:
- Today [Friday] is the last day to add classes.
- Read: §1.2
- Content:
  - the geometric interpretation of the dot product
  - properties of dot products and norms
  - projections
  - the Cauchy-Schwarz-Buniakovsky Inequality
  - the Pythagorean Theorem
  - orthogonal vectors
  - the Triangle Inequality

M:
- Read: Exploration: Vectors and Geometry, p. 29 and §1.3
- Hand in MI1 and exercises HW1:
  - §1.1: 6, 8, 10, 14, 18
  - §1.2: 2, 8, 30_WE, 36, 42, 46, 56_WE, 60
- Content:
  - vector versions of famous (simple) geometric constructions:
    - the midpoint of a line segment
    - the perpendicular bisector of a line segment
  - the normal and general forms of the equation of a line in R²
  - the vector form of the equation of a line in R² or R³
  - parametric equations for a line in R² or R³
W:
- Read: §1.3 (still)
- Content:
  - the normal and general forms of the equation of a plane in R³
  - the vector form of the equation of a plane in R³
  - parametric equations for a plane in R³
  - the distance from a point to a plane in R³
F:
- Read: Exploration: The Cross Product, p. 45-46
- Content:
  - the cross product of two vectors in R³
  - the right-hand rule
  - the geometric interpretation of the cross product

M:
- Read: §2.1
- Hand in MI2 and exercises HW2:
  - §1.3: 2, 10, 14, 18, 20, 22_WE, 24
  - From Exploration: The Cross Product, p.46: 3_WE
  - Review Exercises, p. 56: 8, 10
- Content:
  - a linear equation and systems of linear equations
  - solutions of linear equations and linear systems
  - equivalent linear systems
  - the possible numbers of solutions of a linear system — the geometric viewpoint
  - the coefficient and augmented matrix of a linear system
  - solving a system by back substitution
W:
- Read: §§2.1 (still) & 2.2 (but skip the part about Z_p, on pp.80–82)
- Content:
  - a matrix in row echelon form
  - elementary row operations
  - matrices which are row equivalent
  - Gaussian elimination
  - the rank of a matrix
  - the Rank Theorem
F:
- Read: §2.2 (still; continue to skip the part about Z_p, on pp.80–82)
- Content:
  - a matrix in reduced row echelon form
  - Gaussian-Jordan elimination
  - homogeneous systems

M:

Hand in MI3 and exercises HW3:

§2.1: 6, 18, 22, 28, 32, 34_WE

§2.2: 6, 8, 14, 32, 38, 40_WE

Starting review for Test I

Here is a review sheet for the first midterm test.
W:
- More review for Test I
- Test I.A will take place today, which is the part covering linear systems [the material from §§2.1 & 2.2]
F:
- Test I.B will take place today, on the basics of vectors and applications to lines and planes [the material we covered in Chapter 1]

M:
- Read: §2.3
- going over Test I.A
- Content:
  - the span of some vectors in Rⁿ; a spanning set
  - linear [in]dependence of a set of vectors in Rⁿ
  - theorem relating the consistency of a non-homogeneous system to a condition on the span of the columns of the coefficient matrix
W:
- going over Test I.B
- Read: §2.3 (still)
- Content:
  - linear independence and rank
  - linear independence of m vectors in Rⁿ
F:
- Read: §2.4 [particularly up through p.105]
- Content:
  - generalities on word problems equivalent to linear systems
  - balancing chemical equations, and other situations involving total consumption of limited resources
  - flows in networks

M:
- Read: §3.1 (but skip pp. 143–146, "Partitioned Matrices")
- Content:
  - what is a matrix
  - matrix equality; addition, subtraction, and the zero matrix; scalar multiplication
  - matrix mutiplication: when it is defined, properties, the identity matrix
  - the transpose of a matrix; symmetric matrices
W:
- Hand in revised solutions to Test I.A&B problems for extra credit.
- Read: §3.2
- Content:
  - properties of matrix arithmetic operations, alone and in combination: analogues of the usual associative and distributive laws for numbers continue to apply ...
  - ... but matrix multiplication is not commutative [herein lies the agony and the ecstasy of working with matrices!]
F:
- Read: §3.3
- Hand in MI4 and exercises HW4:
  - §2.2: 39_WE
  - §2.3: 4, 8, 22, 26, 30, 34
  - §2.4: 4_WE, 16_WE, 32_WE
- Content:
  - definition of the inverse of a matrix; an invertible matrix
  - relevance of the inverse for solving linear systems
  - properties of inverses and of the inversion operation

Read: §3.3 (still)
Content:
- revisit what is a matrix inverse, and how knowing one will help solving linear systems
- inverses for 2×2 matrices — when they exist, a formula which computes them
- the Gauss-Jordan method of inverting a matrix

W:

Hand in MI5 and exercises HW5:

§3.1: 10, 12, 16, 18, 22 36_WE

§3.2: 4, 26, 38_WE

§3.3: 4, 12, 22

going over HW4 and HW5

Review for Test II

Here is a review sheet for the second midterm test.

Test II, Takehome Part will be handed out today
F:
- Test II, Takehome Part is due today, in class
- Test II, In-class Part, will take place today

M:
- going over Test II
W:
- Read: §§3.3 & 4.1
- Content:
  - more properties of inverses and of the inversion operation
  - elementary matrices and invertibility
  - the Fundamental Theorem of Invertible Matrices, version 1
  - eigenvalue, eigenvector, and eigenspace
F:
- Hand in revised solutions to Test II problems for extra credit.
- Read: §4.2 (not pp.273(bot)–280)
- Content:
  - simplest examples of computations of eigenstuff — use that λ can be an eigenvalue for a matrix A only if A-λI is non-invertible
  - definition of the determinant of a 2×2 and then 3×3 matrices
  - the Laplace Expansion Formula
  - first properties of determinants

M:
- Read: §4.2 (not pp.273(bot)–280) (still)
- Hand in MI6 and exercises HW6:
  - §3.3: 32, 36, 38, 40_WE
  - §4.1: 2, 6, 8, 12, 24_WE, 37_WE
  - §4.2: 12, 14
- Content:
  - more properties of the determinant
  - Cramer's Rule
W:
- Read: Exploration: Geometric Applications of Determinants, p. 283–288
- Content:
  - determinants and the cross product
  - areas of parallelgrams and volumes of parallelepipeds
  - lines and planes
  - curve fitting
F:
- Read: §4.3 (ignore discussions of dimension and multiplicity)
- Content:
  - the characteristic polynomial char(A) of a square matrix A
  - a method to find eigenstuff with char(A)
  - eigenstuff for triangular matrices
  - the Fundamental Theorem of Invertible Matrices, version 2
  - linear independence of eigenvectors corresponding to disctinct eigenvalues

Last week of meeting for this course.

M:

Hand in MI7 and exercises HW7:

§4.2: 32, 34, 46_WE
From Exploration: Geometric Applications of Determinants, p.283: 1(a)&(b), 6, 11, 12_WE, 15

§4.3: 2(a)&(b), 6(a)&(b), 10(a)&(b), 12(a)&(b), 17_WE, 18_WE,

Final review. Here is a review sheet for the material from Chapters 3 and 4 (so, the material since the second midterm); this should be consulted along side the reviews for Midterm I and Midterm II.
W:
- Final exam, part I This part of the final will focus mostly on the more recent material, from around the time of the second midterm to the end of the course.
F:
- Final exam, part II This part of the final will be more of mini-comprehensive-final, in that it will be fairly evenly divided into problems relating to all parts of the course.

Jonathan Poritz (jonathan.poritz@gmail.com)

Colorado State University, Pueblo; Fall 2011 Math 207 — Matrix and Vector Algebra with Applications Homework Assignments & Course Schedule

Colorado State University, Pueblo; Fall 2011
Math 207 — Matrix and Vector Algebra with Applications
Homework Assignments & Course Schedule