Unit 6 of 10

Systems, Vectors, and Matrices

  • 14 lessons
  • Parts 141 to 168

Many problems involve several unknowns at once. This unit solves linear and nonlinear systems, graphs inequalities in two variables, and uses linear systems to split rational functions into partial fractions, a technique calculus relies on for integration.

It then introduces vectors, first as arrows and then in component form, with the dot product, projections and work. The last part covers matrices, their sums and products, Gaussian elimination, determinants and inverse matrices, all as tools for organizing and solving systems.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Systems of Equations and Inequalities

  1. Linear Systems in Two and Three Variables

    How to solve two or three linear equations at once by substitution or elimination, what the solution means as an intersection, and how to tell one, none or infinitely many solutions apart.

    Theory part 141Practice part 142

  2. Nonlinear Systems and Curve Intersections

    How to find every point where a line meets a parabola or a circle, or where two curves meet, how tangency shows up in the algebra, and why every candidate must be checked.

    Theory part 143Practice part 144

  3. Inequalities in Two Variables

    How to graph an inequality in x and y as a region of the plane, using a boundary curve, solid or dashed lines and test points, and how to combine regions into a feasible region.

    Theory part 145Practice part 146

Partial Fractions

  1. Partial Fraction Decomposition (Linear Factors)

    How to split a rational function whose denominator factors into linear factors into a sum of simple fractions, by dividing first, choosing the right form and solving for the constants.

    Theory part 147Practice part 148

  2. Partial Fraction Decomposition (Quadratic Factors)

    How to decompose a rational function whose denominator has quadratic factors with no real zeros, using linear numerators, one term for each power, and coefficient matching.

    Theory part 149Practice part 150

Vectors

  1. Geometric Vectors and Vector Addition

    What a vector is as an arrow with a magnitude and a direction, why sliding it changes nothing, and how to add vectors head to tail or with the parallelogram law.

    Theory part 151Practice part 152

  2. Algebraic Vectors and Component Form

    How to describe a vector by its horizontal and vertical components, write it with the standard unit vectors i and j, and add, scale and measure vectors with plain arithmetic.

    Theory part 153Practice part 154

  3. The Dot Product and Orthogonality

    How to multiply two vectors to get a number, how that number gives the angle between them, and how it tells at once whether two vectors are perpendicular or parallel.

    Theory part 155Practice part 156

  4. Vector Projections and Work

    How to split a vector into a part along a given direction and a part perpendicular to it, and how the part of a force along the motion gives the work it does, with its units and sign.

    Theory part 157Practice part 158

Matrices

  1. Introduction to Matrices and Matrix Addition

    What a matrix is, how to read its dimensions and entries, when two matrices are equal, and how to add, subtract and scale matrices one entry at a time.

    Theory part 159Practice part 160

  2. Matrix Multiplication

    When two matrices can be multiplied, how each entry of the product is a row-column dot product, why a product means doing one linear map after another, and why the order of the factors matters.

    Theory part 161Practice part 162

  3. Gaussian Elimination and Solution Structure

    How to row reduce an augmented matrix to echelon form, find the pivots and free variables, and read off whether a linear system has one solution, none, or infinitely many.

    Theory part 163Practice part 164

  4. Determinants and Their Geometric Meaning

    How to compute 2 by 2 and 3 by 3 determinants, why a determinant measures how a matrix scales signed area or volume, and how a zero determinant marks a singular matrix.

    Theory part 165Practice part 166

  5. Inverse Matrices and Solving Systems

    When a square matrix has an inverse, how to find it with the 2 by 2 formula or by row reduction, and how an inverse solves a system of linear equations in a single multiplication.

    Theory part 167Practice part 168