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The Vector Space of Polynomials: Span, Linear Independence, and Basis
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Classroom Contents
Linear Algebra
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- 1 What's the big idea of Linear Algebra? **Course Intro**
- 2 What is a Solution to a Linear System? **Intro**
- 3 Visualizing Solutions to Linear Systems - - 2D & 3D Cases Geometrically
- 4 Rewriting a Linear System using Matrix Notation
- 5 Using Elementary Row Operations to Solve Systems of Linear Equations
- 6 Using Elementary Row Operations to simplify a linear system
- 7 Examples with 0, 1, and infinitely many solutions to linear systems
- 8 Row Echelon Form and Reduced Row Echelon Form
- 9 Back Substitution with infinitely many solutions
- 10 What is a vector? Visualizing Vector Addition & Scalar Multiplication
- 11 Introducing Linear Combinations & Span
- 12 How to determine if one vector is in the span of other vectors?
- 13 Matrix-Vector Multiplication and the equation Ax=b
- 14 Matrix-Vector Multiplication Example
- 15 Proving Algebraic Rules in Linear Algebra --- Ex: A(b+c) = Ab +Ac
- 16 The Big Theorem, Part I
- 17 Writing solutions to Ax=b in vector form
- 18 Geometric View on Solutions to Ax=b and Ax=0.
- 19 Three nice properties of homogeneous systems of linear equations
- 20 Linear Dependence and Independence - Geometrically
- 21 Determining Linear Independence vs Linear Dependence
- 22 Making a Math Concept Map | Ex: Linear Independence
- 23 Transformations and Matrix Transformations
- 24 Three examples of Matrix Transformations
- 25 Linear Transformations
- 26 Are Matrix Transformations and Linear Transformation the same? Part I
- 27 Every vector is a linear combination of the same n simple vectors!
- 28 Matrix Transformations are the same thing as Linear Transformations
- 29 Finding the Matrix of a Linear Transformation
- 30 One-to-one, Onto, and the Big Theorem Part II
- 31 The motivation and definition of Matrix Multiplication
- 32 Computing matrix multiplication
- 33 Visualizing Composition of Linear Transformations **aka Matrix Multiplication**
- 34 Elementary Matrices
- 35 You can "invert" matrices to solve equations...sometimes!
- 36 Finding inverses to 2x2 matrices is easy!
- 37 Find the Inverse of a Matrix
- 38 When does a matrix fail to be invertible? Also more "Big Theorem".
- 39 Visualizing Invertible Transformations (plus why we need one-to-one)
- 40 Invertible Matrices correspond with Invertible Transformations **proof**
- 41 Determinants - a "quick" computation to tell if a matrix is invertible
- 42 Determinants can be computed along any row or column - choose the easiest!
- 43 Vector Spaces | Definition & Examples
- 44 The Vector Space of Polynomials: Span, Linear Independence, and Basis
- 45 Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples
- 46 The Span is a Subspace | Proof + Visualization
- 47 The Null Space & Column Space of a Matrix | Algebraically & Geometrically
- 48 The Basis of a Subspace
- 49 Finding a Basis for the Nullspace or Column space of a matrix A
- 50 Finding a basis for Col(A) when A is not in REF form.
- 51 Coordinate Systems From Non-Standard Bases | Definitions + Visualization
- 52 Writing Vectors in a New Coordinate System **Example**
- 53 What Exactly are Grid Lines in Coordinate Systems?
- 54 The Dimension of a Subspace | Definition + First Examples
- 55 Computing Dimension of Null Space & Column Space
- 56 The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!
- 57 Changing Between Two Bases | Derivation + Example
- 58 Visualizing Change Of Basis Dynamically
- 59 Example: Writing a vector in a new basis
- 60 What eigenvalues and eigenvectors mean geometrically
- 61 Using determinants to compute eigenvalues & eigenvectors
- 62 Example: Computing Eigenvalues and Eigenvectors
- 63 A range of possibilities for eigenvalues and eigenvectors
- 64 Diagonal Matrices are Freaking Awesome
- 65 How the Diagonalization Process Works
- 66 Compute large powers of a matrix via diagonalization
- 67 Full Example: Diagonalizing a Matrix
- 68 COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**
- 69 Visualizing Diagonalization & Eigenbases
- 70 Similar matrices have similar properties
- 71 The Similarity Relationship Represents a Change of Basis
- 72 Dot Products and Length
- 73 Distance, Angles, Orthogonality and Pythagoras for vectors
- 74 Orthogonal bases are easy to work with!
- 75 Orthogonal Decomposition Theorem Part 1: Defining the Orthogonal Complement
- 76 The geometric view on orthogonal projections
- 77 Orthogonal Decomposition Theorem Part II
- 78 Proving that orthogonal projections are a form of minimization
- 79 Using Gram-Schmidt to orthogonalize a basis
- 80 Full example: using Gram-Schmidt
- 81 Least Squares Approximations
- 82 Reducing the Least Squares Approximation to solving a system
