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Linear Algebra
A series of linear algebra lectures given in videos by Khan Academy.
Introduction to matrices
Matrix multiplication (part 1)
Matrix multiplication (part 2)
Inverting Matrices (part 1)
Inverting Matrices (part 2)
Inverting Matrices (part 3)
Matrices to solve a system of equations
Matrices to solve a vector combination problem
Singular Matrices
3-variable linear equations
Solving 3 Equations with 3 Unknowns
Introduction to Vectors
Vector Examples
Parametric Representations of Lines
Linear Combinations and Span
Introduction to Linear Independence
More on linear independence
Span and Linear Independence Example
Linear Subspaces
Basis of a Subspace
Vector Dot Product and Vector Length
Proving Vector Dot Product Properties
Proof of the Cauchy-Schwarz Inequality
Linear Algebra: Vector Triangle Inequality
Defining the angle between vectors
Defining a plane in R3 with a point and normal vector
Cross Product Introduction
Proof: Relationship between cross product and sin of angle
Dot and Cross Product Comparison/Intuition
Matrices: Reduced Row Echelon Form 1
Matrices: Reduced Row Echelon Form 2
Matrices: Reduced Row Echelon Form 3
Matrix Vector Products
Introduction to the Null Space of a Matrix
Null Space 2: Calculating the null space of a matrix
Null Space 3: Relation to Linear Independence
Column Space of a Matrix
Null Space and Column Space Basis
Visualizing a Column Space as a Plane in R3
Proof: Any subspace basis has same number of elements
Dimension of the Null Space or Nullity
Dimension of the Column Space or Rank
Showing relation between basis cols and pivot cols
Showing that the candidate basis does span C(A)
A more formal understanding of functions
Vector Transformations
Linear Transformations
Matrix Vector Products as Linear Transformations
Linear Transformations as Matrix Vector Products
Image of a subset under a transformation
im(T): Image of a Transformation
Preimage of a set
Preimage and Kernel Example
Sums and Scalar Multiples of Linear Transformations
More on Matrix Addition and Scalar Multiplication
Linear Transformation Examples: Scaling and Reflections
Linear Transformation Examples: Rotations in R2
Rotation in R3 around the X-axis
Unit Vectors
Introduction to Projections
Expressing a Projection on to a line as a Matrix Vector prod
Compositions of Linear Transformations 1
Compositions of Linear Transformations 2
Linear Algebra: Matrix Product Examples
Matrix Product Associativity
Distributive Property of Matrix Products
Introduction to the inverse of a function
Proof: Invertibility implies a unique solution to f(x)=y
Surjective (onto) and Injective (one-to-one) functions
Relating invertibility to being onto and one-to-one
Determining whether a transformation is onto
Exploring the solution set of Ax=b
Matrix condition for one-to-one transformation
Simplifying conditions for invertibility
Showing that Inverses are Linear
Deriving a method for determining inverses
Example of Finding Matrix Inverse
Formula for 2x2 inverse
3x3 Determinant
nxn Determinant
Determinants along other rows/cols
Rule of Sarrus of Determinants
Determinant when row multiplied by scalar
(correction) scalar muliplication of row
Determinant when row is added
Duplicate Row Determinant
Determinant after row operations
Upper Triangular Determinant
Simpler 4x4 determinant
Determinant and area of a parallelogram
Determinant as Scaling Factor
Transpose of a Matrix
Determinant of Transpose
Transposes of sums and inverses
Transpose of a Vector
Rowspace and Left Nullspace
Visualizations of Left Nullspace and Rowspace
Orthogonal Complements
Rank(A) = Rank(transpose of A)
dim(V) + dim(orthogonoal complelent of V)=n
Representing vectors in Rn using subspace members
Orthogonal Complement of the Orthogonal Complement
Orthogonal Complement of the Nullspace
Unique rowspace solution to Ax=b
Rowspace Solution to Ax=b example
Showing that A-transpose x A is invertible
Projections onto Subspaces
Visualizing a projection onto a plane
A Projection onto a Subspace is a Linear Transformation
Subspace Projection Matrix Example
Another Example of a Projection Matrix
Projection is closest vector in subspace
Least Squares Approximation
Least Squares Examples
Another Least Squares Example
Coordinates with Respect to a Basis
Change of Basis Matrix
Invertible Change of Basis Matrix
Transformation Matrix with Respect to a Basis
Alternate Basis Tranformation Matrix Example
Alternate Basis Tranformation Matrix Example Part 2
Changing coordinate systems to help find a transformation matrix
Introduction to Orthonormal Bases
Coordinates with respect to orthonormal bases
Projections onto subspaces with orthonormal bases
Finding projection onto subspace with orthonormal basis example
Example using orthogonal change-of-basis matrix to find transformation matrix
Orthogonal matrices preserve angles and lengths
The Gram-Schmidt Process
Gram-Schmidt Process Example
Gram-Schmidt example with 3 basis vectors
Introduction to Eigenvalues and Eigenvectors
Proof of formula for determining Eigenvalues
Example solving for the eigenvalues of a 2x2 matrix
Finding Eigenvectors and Eigenspaces example
Eigenvalues of a 3x3 matrix
Eigenvectors and Eigenspaces for a 3x3 matrix
Showing that an eigenbasis makes for good coordinate systems
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