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.

You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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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