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.

**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

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 |

Introduction to Matrices, Complex Numbers,

Matrices, Systems of Linear Equations,

Vectors, Linear Independence and Combinations,

Vector Spaces, Eigenvalues and Eigenvectors

Linear Transformations, Number Sets

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