Syllabus of Algebra and Geometry II

Purpose and intended students

This course is the second half of a journey that begins in Algebra and Geometry I: Numbers, Functions and Plane Geometry. Course I constructed, starting from the Peano axioms, the number systems up to the complex numbers in rectangular form, the algebra of polynomials, real functions and their graphs, the analytic geometry of the plane with its vectors and its conics, and the exponential, logarithmic and hyperbolic functions. Course II takes those objects and carries them into three new territories: trigonometry, which measures angles and turns rotations into a calculation; complex numbers in polar form, where that calculation becomes multiplication; and matrices, which organize linear systems and transformations and which, together with vectors, support the analytic geometry of space with which the course ends.

Its purpose is the same as that of Course I: not only to teach how to calculate, but to show where each calculation comes from. Every result that is used is stated and proved, or is expressly declared admitted when its proof belongs to another course; this is the case, for example, with the fundamental theorem of algebra or with the properties of arc length, which require calculus.

It is intended for anyone who has completed Course I, or who has mastered its contents by another route: students in the last years of secondary school preparing to enter university, first-year university students of science, engineering, economics or mathematics teacher education, and anyone who wishes to rebuild with rigor a mathematics they learned as a collection of recipes. If this is so, the course takes as known the completeness of the real numbers, the study of functions with their inverses and their graph transformations, the line, vectors and the dot product in the plane, the conics, the exponential and the logarithm, and the arithmetic of complex numbers in rectangular form. Whoever is unsure about any of these points can go back to the synthesis of the corresponding unit of Course I and solve its challenges before starting.

Now, the course also occupies a place of transition. It is the prerequisite of Calculus I: Differential Calculus, which differentiates the trigonometric functions starting from the inequality $\sin x < x < \tan x$ proved in unit 1 and which, in turn, leads to Foundations of Mechanics and Calculus II: Integral Calculus, where circular motion, trigonometric substitutions and areas in polar coordinates reappear.

Global learning objectives

The objectives below describe what a student who completes the course should be able to do. Each one is formulated as an observable performance, so that it can be checked with a concrete task; each lesson declares, in turn, its own objectives, and the closing lesson shows, lesson by lesson, to which of these seven each one was assigned. The objectives are numbered so that they can be cited there.

1. Objective 1. Prove statements of trigonometry, of the algebra of complex numbers and of matrices, and of analytic geometry by explicit deductive reasoning, using direct proof, contraposition, proof by contradiction and induction, and distinguish in each proof what is deduced from what is admitted.
2. Objective 2. Apply trigonometry to the measurement of angles and of triangles, study the trigonometric functions and their inverses, prove identities, solve trigonometric equations and inequalities giving their general solution, and describe curves in polar coordinates.
3. Objective 3. Study the geometry of the plane with the tools of the course: calculate angles between lines, eliminate the cross term of a second-degree equation by a rotation of axes and classify the conic it defines, and express rotations, dilations, reflections and projections by means of trigonometry, complex numbers and matrices.
4. Objective 4. Operate with complex numbers in polar and exponential form, calculate their powers and their roots, interpret their operations geometrically and solve polynomial equations over the complex numbers.
5. Objective 5. Solve systems of linear equations with matrices, by means of the Gauss-Jordan method, the rank, determinants and the inverse matrix, and recognize in the number systems, polynomials, matrices and residues modulo $n$ the structures of group, ring and field.
6. Objective 6. Represent points, vectors, lines and planes in space by means of coordinates and equations, operate with the dot product, the cross product and the scalar triple product, and calculate with them angles, distances, areas and volumes, deciding the relative positions of lines and planes by means of linear systems.
7. Objective 7. Model mathematically situations from science, technology and everyday life with the tools of the course, and communicate in writing the setups, the calculations and the proofs with correct notation, complete argumentation and a check of the plausibility of the results.

Objectives 2 to 6 correspond, in essence, to the major topics of the course: objective 2 to units 1 and 2, objective 4 to unit 3, objective 5 to unit 4 and objective 6 to unit 5, while objective 3 gathers a thread that runs through units 1 to 4, the geometry of the plane seen successively with sines and cosines, with complex numbers and with matrices. The first and the seventh, by contrast, are transversal: they do not belong to one unit, but are exercised in all of them. From this it follows that their achievement cannot be attributed to a particular lesson, and that they should be cultivated throughout the whole course. Now, both also have lessons that serve them principally: the first, the syntheses, whose central portion reconstructs the deductive chain of each unit; the seventh, the lessons in which setting up problems is the very subject matter, the one on the angle between lines and its applications and the one on linear systems.

Contents

The course is organized in seven units. The first and the last frame the journey; the five intermediate ones develop the contents. Each of these opens with an introduction lesson, which presents its lessons, the historical background of its ideas, the problem that gave rise to them and their relevance, and closes with a synthesis lesson, which reconstructs the deductive chain of the unit, connects it with the rest of the course and with Course I, and proposes ten challenges of greater difficulty than the problems of the lessons.

Unit 0. Course overview

It consists of this lesson, which sets out the purpose, the global objectives, the contents, the workload and the design of the course.

Unit 1. Trigonometric functions and polar coordinates

Its objective is to define the trigonometric functions on the unit circle, study their graphs and their inverses, solve triangles and describe curves in polar coordinates.

1. Trigonometric functions and polar coordinates: introduction.
2. Angles and their measure: degrees and radians.
3. Trigonometric ratios and the unit circle.
4. The trigonometric functions and their graphs.
5. The inverse trigonometric functions.
6. Solving triangles: the laws of sines and cosines.
7. Polar coordinates and polar curves.
8. Trigonometry and polar coordinates: synthesis and challenges.

Unit 2. Analytic trigonometry

Its objective is to turn the trigonometric functions into an algebraic calculus, with identities, addition formulas, equations and inequalities, and to apply it to the angles between lines and to conics with oblique axes.

1. Analytic trigonometry: introduction.
2. Fundamental trigonometric identities.
3. Sum and difference formulas, double angle and half angle.
4. Sum-to-product formulas and the harmonic form.
5. Trigonometric equations and inequalities.
6. The angle between two lines and applications of trigonometry.
7. Rotation of axes and the general second-degree equation.
8. Analytic trigonometry: synthesis and challenges.

Unit 3. Complex numbers in polar form

Its objective is to describe each complex number by its modulus and its argument, so that the product reads as a rotation combined with a dilation, powers as multiples of an angle and roots as regular polygons, and to solve polynomial equations with this. It takes up the complex numbers in rectangular form constructed in unit 2 of Course I.

1. Complex numbers in polar form: introduction.
2. The polar form of complex numbers.
3. De Moivre’s formula and powers of complex numbers.
4. The nth roots of a complex number and the roots of unity.
5. The complex exponential and Euler’s formula.
6. Geometry with complex numbers.
7. Complex polynomial equations: the Cardano and Ferrari formulas.
8. Complex numbers in polar form: synthesis and challenges.

Unit 4. Matrices and algebraic structures

Its objective is to make the table of coefficients of a linear system an object with operations of its own, to solve the systems with it, to read matrices as transformations and to condense what is common to the calculations of the two courses into the notions of group, ring and field.

1. Matrices and algebraic structures: introduction.
2. Matrices and matrix operations.
3. Linear systems and the Gauss-Jordan method.
4. Determinants: properties and Cramer’s rule.
5. The inverse matrix.
6. Matrices as transformations of the plane.
7. Complex numbers as matrices.
8. Algebraic structures: groups, rings and fields.
9. Matrices and algebraic structures: synthesis and challenges.

Unit 5. Analytic geometry of space

Its objective is to extend to space the analytic geometry of the plane studied in unit 4 of Course I: to describe points and vectors with three coordinates, starting from an explicit list of admitted facts of the Euclidean geometry of space; to measure lengths and angles with the dot product, proving that the angle so defined is that of elementary geometry; to obtain from the third-order determinant the cross product and the scalar triple product, with their reading as an area, as a volume and as a criterion of coplanarity; to write lines and planes by means of equations and decide their relative positions with the rank and the Rouché-Capelli theorem (known in the Spanish-speaking world as the Rouché-Frobenius theorem) of unit 4; and to calculate distances, angles, projections and the common perpendicular of two skew lines, with applications to mechanics, crystallography, air navigation and satellite positioning. The unit studies the sphere, but not the other quadric surfaces (ellipsoids, paraboloids, hyperboloids, second-degree cones and cylinders), which are not treated in this course; in the current organization of the site, their study belongs to Calculus III.

1. Analytic geometry of space: introduction.
2. Vectors in space and the dot product.
3. The cross product and the scalar triple product.
4. Lines and planes in space.
5. Distances, angles and volumes in space.
6. Analytic geometry of space: synthesis and challenges.

Unit 6. Course closing

It consists of the lesson Final review of Algebra and Geometry II, which recalls the seven global objectives and shows, in a figure, to which of them each lesson of the course was assigned and which specific objectives each unit achieved, and gathers by objective the first attempt at the exit assessments.

Workload and credits

The workload of a course is not measured by hours of lecturing, which do not exist in an asynchronous course, but by the student’s total work: reading, following the proofs, solving problems and reviewing. The estimate counts the reading time of the text of each lesson, twenty minutes per solved problem, thirty-five per proposed problem, forty-five per challenge and twelve per entry or exit assessment, and adds a margin of one tenth for review.

Results

Taken together, the lessons of units 1 to 5 contain $145$ solved problems, $145$ proposed problems and $50$ challenges. The estimated workload per unit, rounded to the hour, is as follows:

• Unit 0, course overview: $1$ hour.
• Unit 1, trigonometric functions and polar coordinates: $46$ hours.
• Unit 2, analytic trigonometry: $46$ hours.
• Unit 3, complex numbers in polar form: $46$ hours.
• Unit 4, matrices and algebraic structures: $54$ hours.
• Unit 5, analytic geometry of space: $36$ hours.
• Unit 6, course closing: less than one hour.

Unit 5 is the shortest in number of lessons, but its four content lessons are among the longest of the course. The total is about $230$ hours of student work. This figure should be read for what it is: an estimate for a student who studies all the proofs and solves all the problems and challenges. Whoever already masters some of the contents, or restricts themselves to the proposed problems without tackling the challenges, will need considerably less; whoever studies the subject for the first time and without the habit of proving may need more.

Now, the figure of $230$ hours is nominal: it is the one that sets the credits of the course, and it assumes reading the text at a constant pace and three quarters of an hour per challenge. Until it is measured with real students, it should be accompanied by a realistic range. If reading proceeded at half the assumed pace, as happens when each proof is rewritten with pencil and paper, and each challenge required seventy-five minutes instead of forty-five, the same method would give about $298$ hours. Therefore, the workload to be expected lies between about $230$ and about $300$ hours. The upper end is a prudent bound, not a measurement; the range will be corrected once the time spent by those who study the course is known, and until then the credits are calculated with the nominal figure.

Suggested pace

To name the paces, this plan and that of Course I use the same criterion: full-time study is that of an academic year of $60$ credits, about $1650$ hours spread over some forty weeks, that is, about $40$ hours a week, and half-time is half of that, about $20$ hours a week. From this follows a recommended pace of one sixteen-week semester, with about $14$ hours of study a week, somewhat more than one third of full-time study and, therefore, less than half-time; with the realistic range, from $14$ to $19$ hours a week, it approaches half-time without reaching it. Thus Courses I and II occupy two consecutive semesters. A more relaxed pace, of about $7$ hours a week, spreads the course over two semesters; an intensive one, of about $29$ hours a week, somewhat less than three quarters of full-time study, completes it in eight weeks, which is equivalent to the workload of a student who is taking no other demanding subject in the same period.

Credits

The credits of this site measure the total workload that a course requires of the person studying it (reading, problems, challenges and review), not class hours. One credit is equivalent to about $27.5$ hours of work, and a year of full-time study to $60$ credits, that is, to about $1650$ hours; this is the same criterion with which the preceding subsection names the paces of study.

Now, dividing the estimated $230$ hours by $27.5$ gives 8.4, which is rounded to $8$. If a band of $25$ to $30$ hours of work is allowed for the credit, the number $C$ of credits of the course is bounded between 7.7 and 9.2:

Therefore, the course has $8$ credits, somewhat more than one eighth of a year of full-time study. Anyone wishing to study only a portion of the course can use the list of hours per unit: units 0 to 4, for example, add up to about $193$ hours, that is, about $7$ credits.

Their use is practical: they make it possible to compare the workload of the courses on the site with one another and to decide how many to study at the same time without exceeding the time available. Someone who can devote half-time study, for example, to a sixteen-week semester, that is, about $320$ hours, can take on courses adding up to about $12$ credits; with full-time study over the same semester, about $23$.

These credits are a measure specific to this site, for planning study: they do not correspond to any official credit system, nor do they entail academic recognition by any institution.

Course design

Mode of study

The course is asynchronous, online and self-paced: it has no dates, live sessions or deadlines, and each student advances at the speed that their preparation and their time allow. It is offered in the fourteen languages of the site, with the same mathematics in all of them: the formulas and the steps of the proofs are identical in each version, and only the text is translated. The lessons declare which other lessons it is advisable to have studied beforehand, including those of Course I on which they depend, so that the recommended order is explicit and can be verified.

The structure of a lesson

All topical lessons follow the same structure. They begin with a summary and a list of their own learning objectives, formulated as observable performances. Then comes the theory, with the definitions, the theorems stated precisely and their proofs, and the examples. The proofs and the examples are presented as step-by-step players: each step writes one line of the reasoning, highlights the terms involved and, when it relies on an earlier result, goes back to it and highlights it as it is read, so that the memory of the reasoning stays in view; many are accompanied by a figure that changes with each step. The lesson closes with five solved problems, with their step-by-step solution, and five proposed problems, each with a folded hint and, after it, the final answer, so that the student can check their result without the path being revealed; the proposed problems that ask for a proof add, folded after the answer, a worked solution, which should be opened only after writing one’s own.

The introduction and synthesis lessons have a structure of their own. The introduction presents the unit, its history, the problem that gave rise to it and its relevance; the synthesis reconstructs in a figure the deductive chain of the unit, with the results it reuses and the links it proves, connects it with the earlier units and with Course I, and proposes ten challenges, each with a hint, a final answer and a folded worked solution, which combine several lessons and demand more invention than the proposed problems.

The voice reader

Every lesson can be listened to aloud with the site’s reader, which uses the device’s own voices: the text is read just as it is written and the formulas are pronounced in words. This makes it possible to review a lesson away from the screen, to follow a proof by listening to it while watching the step-by-step player, and to study with visual impairments.

Assessment

The course neither grades nor certifies: its assessment on the site is formative. The proposed problems make it possible to check the achievement of the objectives of each lesson, and the challenges, that of the objectives of each unit. Each of the five units also has an entry assessment in its introduction and an exit assessment in its synthesis, before the challenges: short self-assessments, corrected in the browser without grading; the entry assessment has nine or ten questions about the unit, plus between two and six about what the unit takes as known from the earlier ones or from Course I, and the exit assessment, eleven or twelve; both indicate which lesson to review after each error and are compared with each other to show progress. The final review gathers the first attempt at each exit assessment into a profile by global objective, which is stored, like them, only in the student’s browser. The validation of skills, with certification value, will be done later and outside this site, on another platform.

In addition, the following evidence of achievement is suggested for the global objectives, which each student can produce on their own:

• Objective 1: write, without consulting the lesson, the proof of a theorem of the course, such as the law of cosines, De Moivre’s formula or the uniqueness of the inverse matrix.
• Objective 2: prove a nontrivial trigonometric identity and solve a trigonometric equation giving its general solution.
• Objective 3: classify the conic defined by a second-degree equation with a cross term, by rotating the axes, and express a rotation of the plane as a product by a complex number and as a matrix.
• Objective 4: calculate the $n$th roots of a complex number, represent them in the plane and solve a cubic equation with Cardano’s formula.
• Objective 5: solve a linear system with a parameter, analyzing the cases by means of the rank, and decide whether a given structure is a group, a ring or a field.
• Objective 6: find the relative position of two lines in space and the distance between them, and the volume of a tetrahedron given by its vertices.
• Objective 7: write out the complete solution of a modeling problem, with setup, working, answer and plausibility check, so that another person can follow it without additional explanations.

How to study this course

The course is designed to be studied in order. Each unit presupposes the earlier ones and Course I, and many proofs cite results from earlier lessons, of one course or the other; skipping a lesson forces one, sooner or later, to go back to it.

Each unit has two checkpoints. The entry assessment, at the beginning of the introduction, is the starting point: it should be answered before studying the unit, without consulting anything, noting the lessons to which the errors refer, which are the ones that call for more care. The exit assessment, in the synthesis and before the challenges, is the contrast: it is answered after finishing the lessons, also without consulting, and is compared with the entry assessment to see what changed and which lessons should be reviewed. In both, the first attempt is what counts, being the one that is compared and the one that the profile of the final review gathers; later attempts are made with the feedback already seen. The challenges complete the check, because they require combining and writing out what the short questions only touch on. From this it follows that whoever believes they already master a unit should not decide it by a single test: if their exit assessment is solid and they solve the challenges of the synthesis, they may move on to the following unit; if not, it is advisable to go through the lessons that the one and the others point to.

In each lesson, it is advisable first to read the summary and the objectives, which say what is sought; then to study the theory with pencil and paper, trying to anticipate each step of the proofs before the player writes it; to attempt the solved problems before looking at their solution, and to solve the proposed ones without help, opening the hint only when the attempt has been exhausted, and the answer only at the end, to check. A problem whose answer was consulted before attempting it has not been solved, but read.

The challenges are harder than the problems of the lessons, and it is normal for some to require several attempts on different days. It is not advisable to dwell indefinitely on one: noting it down, continuing to advance and coming back to it later is often the shortest path. Now, the estimated workload assumes that all of them are attempted seriously, and in that effort much of what the course can teach is at stake.

Finally, it is advisable to write. Writing out a proof or a complete solution, with its premises and its conclusions, is the only way to know whether one has understood, and it is also the seventh of the course objectives. A study notebook, in which the definitions and the theorems of each unit are rewritten in one’s own words, is the best review material the course can leave behind, and it will be all the more so in Calculus I, which will return to almost everything that is proved here.