Kahn - levandosky
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Khan VideoTopic (1-16)da shit
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Introduction to matrices
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Matrix multiplication (part 1)
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Matrix multiplication (part 2)
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Idea Behind Inverting a 2x2 Matrix
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Inverting matrices (part 2)
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Inverting Matrices (part 3)
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Matrices to solve a system of equations
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Matrices to solve a vector combination problem
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Singular Matrices
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3-variable linear equations (part 1)
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Solving 3 Equations with 3 Unknowns
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Introduction to Vectors
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Vector Examples
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Parametric Representations of Lines
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Linear Combinations and Span
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Introduction to Linear Independence
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More on linear independence
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Span and Linear Independence Example
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Linear Subspaces
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Basis of a Subspace
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Vector Dot Product and Vector Length
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Proving Vector Dot Product Properties
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Proof of the Cauchy-Schwarz Inequality
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Vector Triangle Inequality
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Defining the angle between vectors
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Defining a plane in R3 with a point and normal vector
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Cross Product Introduction
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Proof: Relationship between cross product and sin of angle
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Dot and Cross Product Comparison/Intuition
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Matrices: Reduced Row Echelon Form 1
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Matrices: Reduced Row Echelon Form 2
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Matrices: Reduced Row Echelon Form 3
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Matrix Vector Products
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Introduction to the Null Space of a Matrix
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Null Space 2: Calculating the null space of a matrix
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Null Space 3: Relation to Linear Independence
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Column Space of a Matrix
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Null Space and Column Space Basis
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Visualizing a Column Space as a Plane in R3
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Proof: Any subspace basis has same number of elements
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Dimension of the Null Space or Nullity
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Dimension of the Column Space or Rank
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Showing relation between basis cols and pivot cols
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Showing that the candidate basis does span C(A)
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A more formal understanding of functions
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Vector Transformations
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Linear Transformations
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Matrix Vector Products as Linear Transformations
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Linear Transformations as Matrix Vector Products
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Image of a subset under a transformation
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im(T): Image of a Transformation
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Preimage of a set
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Preimage and Kernel Example
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Sums and Scalar Multiples of Linear Transformations
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More on Matrix Addition and Scalar Multiplication
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Linear Transformation Examples: Scaling and Reflections
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Linear Transformation Examples: Rotations in R2
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Rotation in R3 around the X-axis
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Unit Vectors
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Introduction to Projections
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Expressing a Projection on to a line as a Matrix Vector prod
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Compositions of Linear Transformations 1
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Compositions of Linear Transformations 2
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Matrix Product Examples
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Matrix Product Associativity
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Distributive Property of Matrix Products
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Introduction to the inverse of a function
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Proof: Invertibility implies a unique solution to f(x)=y
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Surjective (onto) and Injective (one-to-one) functions
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Relating invertibility to being onto and one-to-one
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Determining whether a transformation is onto
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Exploring the solution set of Ax=b
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Matrix condition for one-to-one trans
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Simplifying conditions for invertibility
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Showing that Inverses are Linear
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Deriving a method for determining inverses
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Example of Finding Matrix Inverse
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Formula for 2x2 inverse
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3x3 Determinant
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nxn Determinant
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Determinants along other rows/cols
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Rule of Sarrus of Determinants
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Determinant when row multiplied by scalar
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(correction) scalar multiplication of row
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Duplicate Row Determinant
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Determinant after row operations
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Upper Triangular Determinant
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Simpler 4x4 determinant
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Determinant and area of a parallelogram
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Determinant as Scaling Factor
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Transpose of a Matrix
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Determinant of Transpose
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Transpose of a Matrix Product
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Transposes of sums and inverses
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Transpose of a Vector
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Rowspace and Left Nullspace
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Visualizations of Left Nullspace and Rowspace
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Orthogonal Complements