Linear Algebra

Vectors and Matrices

This section introduces the algebraic structure of vectors and matrices, foundational for solving systems of linear equations and for applications in calculus and geometry.

• Vector: An n-dimensional vector is an ordered n-tuple of real numbers, .

• Scalar: A real number, .

• Vector Operations:

  • Addition:

  • Scalar Multiplication:

  • Dot Product:

• Matrix: An array of real numbers, .

• Matrix Operations:

  • Addition:

  • Scalar Multiplication:

  • Matrix Multiplication:

• Identity Matrix: has 1's on the diagonal, 0's elsewhere.

Example: For and , , .

Systems of Linear Equations

Systems of equations can be represented and solved using matrices and vectors.

• System: , where is the coefficient matrix, the variable vector, the constant vector.

• Solution: is a solution if .

• Augmented Matrix: combines coefficients and constants for row operations.

• Row Echelon Form: Used to simplify systems for solution via Gaussian elimination.

• Reduced Row Echelon Form: Unique form for a system, found via Gauss-Jordan elimination.

• Pivot Variables: Variables corresponding to leading 1's in row echelon form; others are free variables.

Example: Solve by writing the augmented matrix and applying row operations.

Inverse Matrices and Determinant

Square matrices may have inverses, useful for solving when is invertible.

• Invertible Matrix: is invertible if such that .

• Determinant (2x2): for .

• Inverse (2x2): , if .

Example: For , , so is invertible.

Geometry of 3D Space

Vectors in 3D Space

Vectors in are used to represent points, directions, and displacements in space.

• Norm (Magnitude):

• Unit Vector:

• Standard Basis: , ,

Example: The vector from to is .

Distance and Surfaces in 3D

• Distance:

• Plane: for normal vector and point

• Sphere:

• Cylinder: (parallel to z-axis)

The Dot Product

• Dot Product:

• Orthogonality:

• Projection:

Example: The angle between and is since .

The Cross Product

• Cross Product:

• Magnitude:

• Geometric Meaning: is perpendicular to both and

• Area: Parallelogram area = ; triangle area =

Triple Products and Volume

• Scalar Triple Product: gives the volume of the parallelepiped formed by

• Vector Triple Product:

Applications of Differentiation

Sketching of Curves

Derivatives provide information about the shape and features of function graphs.

• Increasing/Decreasing: increasing; decreasing

• Critical Point: Where or undefined

• First Derivative Test: Determines local maxima/minima by sign changes of

• Second Derivative Test: local min; local max

• Concavity: (concave up), (concave down)

• Inflection Point: Where concavity changes

Optimisation

• Global Extreme Values: is a global max/min if or for all in domain

• Extreme Value Theorem: A continuous function on attains both a maximum and minimum

• Closed Interval Method: Evaluate at critical points and endpoints to find global extrema

Taylor Series

• Power Series:

• Taylor Series:

• Maclaurin Series: Taylor series at

• Taylor Polynomial:

• Remainder Term:

• Taylor's Inequality: , where bounds

Definite Integrals and the Fundamental Theorem of Calculus

Riemann Sums and Definite Integrals

• Definite Integral:

• Area Under a Curve: for

• Integrable Function: Bounded and continuous except for finitely many discontinuities

Properties of Definite Integrals

• Linearity:

• Interval Additivity:

• Comparison: If on , then

Mean Value Theorems

• For Derivatives: for some

• For Integrals: such that

The Fundamental Theorem of Calculus (FTC)

• Part 1: If , then

• Part 2: , where is any antiderivative of

Substitution Rule and Integration by Parts

• Substitution:

• Integration by Parts:

Applications of Integration

Area Between Curves

• Area between and on is

Volume of Solids

• Disk Method:

• Washer Method:

• Cylindrical Shells:

Hyperbolic and Inverse Hyperbolic Functions

Hyperbolic Functions

• Identities:

Inverse Hyperbolic Functions

• ,

• ,

Derivatives and Integrals

Integration Techniques

Trigonometric Integrals

• Use identities to reduce powers and integrate products like

• For , use

Trigonometric Substitution

• For , use

• For , use

• For , use

Integration of Rational Functions

• Decompose into partial fractions for integration

• For , if , use long division first

• Decompose into terms like or

Further Applications of Integration

Arc Length

• For , arc length on is

Area of a Surface of Revolution

• Rotating about the x-axis:

Combinatorics

Counting Principles

• Product Rule: ways for A, ways for B ways for both

• Sum Rule: ways for A, ways for B ways for either

Permutations and Combinations

• Permutation: (order matters)

• Combination: (order does not matter)

Binomial Theorem

• For real exponents: ,

*Additional info: The above notes are structured to cover all main topics and subtopics as outlined in the MTHS121 study guide, with expanded academic context, definitions, formulas, and examples for each major area of Calculus II. For full mastery, students should supplement these notes with textbook exercises and proofs as indicated in the guide.*