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

Linear Algebra Calculator with step by step solutions
Introduction to Matrices, Complex Numbers,
Matrices, Systems of Linear Equations,
Vectors, Linear Independence and Combinations,
Vector Spaces, Eigenvalues and Eigenvectors
Linear Transformations, Number Sets

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.