Lesson 79 of 84
Proving that orthogonal projections are a form of minimization
Lessons84 lessons
- 1What's the big idea of Linear Algebra? **Course Intro**
- 2What is a Solution to a Linear System? **Intro**
- 3Visualizing Solutions to Linear Systems - - 2D & 3D Cases Geometrically
- 4Rewriting a Linear System using Matrix Notation
- 5Using Elementary Row Operations to Solve Systems of Linear Equations
- 6Using Elementary Row Operations to simplify a linear system
- 7Examples with 0, 1, and infinitely many solutions to linear systems
- 8Row Echelon Form and Reduced Row Echelon Form
- 9Back Substitution with infinitely many solutions
- 10The Gaussian Algorithm Visualized
- 11What is a vector? Visualizing Vector Addition & Scalar Multiplication
- 12Introducing Linear Combinations & Span
- 13How to determine if one vector is in the span of other vectors?
- 14Matrix-Vector Multiplication and the equation Ax=b
- 15Matrix-Vector Multiplication Example
- 16Proving Algebraic Rules in Linear Algebra --- Ex: A(b+c) = Ab +Ac
- 17The Big Theorem, Part I
- 18Writing solutions to Ax=b in vector form
- 19Geometric View on Solutions to Ax=b and Ax=0.
- 20Three nice properties of homogeneous systems of linear equations
- 21Linear Dependence and Independence - Geometrically
- 22Determining Linear Independence vs Linear Dependence
- 23Making a Math Concept Map | Ex: Linear Independence
- 24Transformations and Matrix Transformations
- 25Three examples of Matrix Transformations
- 26Linear Transformations
- 27Are Matrix Transformations and Linear Transformation the same? Part I
- 28Every vector is a linear combination of the same n simple vectors!
- 29Matrix Transformations are the same thing as Linear Transformations
- 30Finding the Matrix of a Linear Transformation
- 31One-to-one, Onto, and the Big Theorem Part II
- 32The motivation and definition of Matrix Multiplication
- 33Computing matrix multiplication
- 34Visualizing Composition of Linear Transformations **aka Matrix Multiplication**
- 35Elementary Matrices
- 36You can "invert" matrices to solve equations...sometimes!
- 37Finding inverses to 2x2 matrices is easy!
- 38Find the Inverse of a Matrix
- 39When does a matrix fail to be invertible? Also more "Big Theorem".
- 40Visualizing Invertible Transformations (plus why we need one-to-one)
- 41Invertible Matrices correspond with Invertible Transformations **proof**
- 42Determinants - a "quick" computation to tell if a matrix is invertible
- 43Determinants can be computed along any row or column - choose the easiest!
- 44Vector Spaces | Definition & Examples
- 45The Vector Space of Polynomials: Span, Linear Independence, and Basis
- 46Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples
- 47The Span is a Subspace | Proof + Visualization
- 48The Null Space & Column Space of a Matrix | Algebraically & Geometrically
- 49The Basis of a Subspace
- 50Finding a Basis for the Nullspace or Column space of a matrix A
- 51Finding a basis for Col(A) when A is not in REF form.
- 52Coordinate Systems From Non-Standard Bases | Definitions + Visualization
- 53Writing Vectors in a New Coordinate System **Example**
- 54What Exactly are Grid Lines in Coordinate Systems?
- 55The Dimension of a Subspace | Definition + First Examples
- 56Computing Dimension of Null Space & Column Space
- 57The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!
- 58Changing Between Two Bases | Derivation + Example
- 59Visualizing Change Of Basis Dynamically
- 60Example: Writing a vector in a new basis
- 61What eigenvalues and eigenvectors mean geometrically
- 62Using determinants to compute eigenvalues & eigenvectors
- 63Example: Computing Eigenvalues and Eigenvectors
- 64A range of possibilities for eigenvalues and eigenvectors
- 65Diagonal Matrices are Freaking Awesome
- 66How the Diagonalization Process Works
- 67Compute large powers of a matrix via diagonalization
- 68Full Example: Diagonalizing a Matrix
- 69COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**
- 70Visualizing Diagonalization & Eigenbases
- 71Similar matrices have similar properties
- 72The Similarity Relationship Represents a Change of Basis
- 73Dot Products and Length
- 74Distance, Angles, Orthogonality and Pythagoras for vectors
- 75Orthogonal bases are easy to work with!
- 76Orthogonal Decomposition Theorem Part 1: Defining the Orthogonal Complement
- 77The geometric view on orthogonal projections
- 78Orthogonal Decomposition Theorem Part II
- 79Proving that orthogonal projections are a form of minimization
- 80Using Gram-Schmidt to orthogonalize a basis
- 81Full example: using Gram-Schmidt
- 82Rotation Matrices || Linear Algebra Fundamentals
- 83Least Squares Approximations
- 84Reducing the Least Squares Approximation to solving a system