Course Outline
1. Introduction to Calculus
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1.1 Limits and Continuity
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1.2 Derivatives
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1.3 Applications of Derivatives
2. Integral Calculus
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2.1 Antiderivatives
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2.2 Definite Integrals
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2.3 Applications of Integration
3. Calculus & Linear Algebra
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3.1 Vectors and Vector Spaces
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3.2 Matrices and Linear Transformations
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3.3 Eigenvalues and Eigenvectors
4. Multivariable Calculus
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4.1 Partial Derivatives
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4.2 Multiple Integrals
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4.3 Vector Calculus
5. Differential Equations
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5.1 First-Order Equations
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5.2 Second-Order Equations
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5.3 Applications
Vectors and Vector Spaces
Module 3.1 - Calculus & Linear Algebra
Estimated Time
45 minutes
Next Milestone
Matrices
Learning Objectives
Understand the concept of vectors in n-dimensional space
Perform vector operations (addition, scalar multiplication)
Calculate dot products and cross products
Identify vector spaces and their properties
Introduction to Vectors and Vector Spaces
Prof. Elizabeth Carter explains the fundamental concepts of vectors, their operations, and the properties of vector spaces.
Key Concepts
1. Vector Definition
A vector is a mathematical object that has both magnitude and direction. In n-dimensional space, a vector can be represented as an ordered list of n numbers.
v = (v₁, v₂, ..., vₙ)
2. Vector Operations
Basic operations with vectors include addition, subtraction, and scalar multiplication.
Addition: u + v = (u₁ + v₁, u₂ + v₂, ..., uₙ + vₙ)
Subtraction: u - v = (u₁ - v₁, u₂ - v₂, ..., uₙ - vₙ)
Scalar Multiplication: c·v = (c·v₁, c·v₂, ..., c·vₙ)
3. Dot Product
The dot product of two vectors is a scalar value that represents the product of their magnitudes and the cosine of the angle between them.
u · v = u₁v₁ + u₂v₂ + ... + uₙvₙ
u · v = |u||v|cos(θ)
4. Vector Spaces
A vector space is a set of vectors that is closed under vector addition and scalar multiplication, satisfying specific axioms.
Key Properties:
- Closure under addition: u + v ∈ V for all u, v ∈ V
- Closure under scalar multiplication: c·v ∈ V for all v ∈ V and scalar c
- Associativity, commutativity, distributivity, etc.
Interactive Example: Vector Addition
Practice Problems
Problem 1: Vector Operations
Given vectors u = (3, -1, 4) and v = (2, 5, 0), calculate:
a) u + v
b) 2u - v
c) u · v (dot product)
Problem 2: Vector Space Properties
Determine whether the following set forms a vector space. Justify your answer.
The set of all vectors (x, y, z) where x + y + z = 0.
Downloadable Resources
Lecture Notes: Vectors and Vector Spaces
PDF, 2.4 MB
Vector Calculator Spreadsheet
XLSX, 1.1 MB
Vector Spaces Presentation Slides
PPTX, 3.8 MB
Practice Problem Set with Solutions
PDF, 1.7 MB
AI Tutor Support
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