What is a vector? Visualizing Vector Addition & Scalar Multiplication

ক্লাস84টি ক্লাস

• ✓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

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