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Linear Algebra




 
Looking for free Linear Algebra help?
We have a series of linear algebra lectures given in videos by Khan Academy.
In this series, we will learn matrices, vectors, vector spaces, determinants and transformations.

Introduction to matrices Matrix multiplication Inverting Matrices (part 1) Inverting Matrices (parts 2 & 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 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 Reduced Row Echelon Form 2
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 product
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 multiplication 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 Product
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(orthogonal complement 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
Projection is closest vector in subspace
Least Squares Approximation Least Squares Examples
Coordinates with Respect to a Basis


 
Change of Basis Matrix Invertible Change of Basis Matrix
Transformation Matrix with Respect to a Basis
Alternate Basis Transformation Matrix Example
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
 

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