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▾Linear algebra
- •Adding and subtracting matrices
- •Adding vectors in polar form
- •Applications of linear algebra
- •Applications of vectors
- •Area of a parallelogram using determinant
- •Area of a triangle using determinant
- •Are the vectors parallel, orthogonal, or neither?
- ▸Basic transformation matrices
  - •Transforming a point
  - •Composition of transformations
- ▸Basis
  - •Basis for a null space
  - •Basis for a span
- ▸Cramer's rule
  - •Solving linear systems in two variables using Cramer's rule
  - •Solving linear systems in three variables using Cramer's rule
- •Definition of \(\mathbb{R}^n\)
- ▸Determinants
  - •Determinant using row echelon form
  - •Determinant using cofactors
  - ▸Determinant using row operations
    - •Determinant of a \(4{\times}4\) matrix using row operations
  - ▸Determinant of a \(3{\times}3\) matrix
    - •Determinant of a \(3{\times}3\) matrix using expansion by minors
    - •Determinant of a \(3{\times}3\) matrix using the diagonal method
  - •Determinant of a \(2{\times}2\) matrix
  - •Area of a triangle using determinants
  - •Intuition: Determinants
- •Diagonalization
- ▸Eigenvalues and eigenvectors
  - •Power method
  - •Shifted inverse power method
  - •Finding the eigenvalues and eigenvectors of \(2{\times}2\) matrices
  - •Finding the eigenvalues and eigenvectors of \(3{\times}3\) matrices
  - •Finding the eigenvalues and eigenvectors of \(4{\times}4\) matrices
- •Elementary matrices
- •Applications of the dot product
- •Gaussian elimination
- ▸Geometric proofs using vectors
  - •Proof: Law of cosines using the dot product
  - •Proof: Rhombus diagonals are perpendicular bisectors (linear algebra)
  - •Proof: Varignon's theorem (linear algebra)
  - •Proof: Thales's theorem using the dot product
- •Gershgorin circle theorem
- •History
- •Length of a vector
- •Linear combinations
- ▸Linear transformations
  - •Linear transformations on \(\mathbb{R}^2\)
- ▸Magnitude and direction form of vectors
  - •Lesson
  - •Practice
- •Matrix inverses
- ▸Matrix multiplication
  - •Intuition: Matrix multiplication
- ▸Multiplying a matrix by a scalar
  - •Practice
- •Multiplying a vector by a scalar
- •Orthogonal vectors
- •Parallel vectors
- •Proof: Cauchy–Schwarz inequality
- •Proof: Cross product distributes over addition
- •Proof: Dot product formula
- •Proof: Vector addition is commutative and associative
- ▸Proof the cross product is distributive
  - •Algebraic proof the cross product is distributive
  - •Geometric proof the cross product is distributive
- •Properties of matrix multiplication
- •Properties of scalar multiplication
- •Section formula
- ▸Solving linear systems of equations
  - ▸Solving linear systems of equations using matrix inverses
    - •Solving linear systems of equations in two variables using matrix inverses
  - •Solving linear systems of equations in three variables using Gaussian elimination
  - •Consistency of a system of linear equations
- •Gauss–Jordan elimination
- •Solving linear systems of equations in two variables using Gauss–Jordan elimination
- •Solving matrix equations
- •Transpose and its properties
- ▸Unit vectors
  - •Finding the unit vector
- •Vector magnitude
- ▸Vector projection
  - •Proof: Vector projection formula
- ▸Vectors
  - ▸3D vectors
    - •3D vector operations
  - ▸Adding and subtracting vectors
    - •Adding vectors
    - •Subtracting vectors
  - ▸Components of vectors
    - •Lesson
    - •Practice
  - •Equivalent vectors
  - •Properties of vector addition
  - •Properties of vector addition and scalar multiplication
- •What is an orthogonal set?
- •What is a vector space?
- •Zero matrix
- •William Rowan Hamilton
- ▸Solving linear systems of equations in three variables
  - •Solving linear systems of equations in three variables (word problems)
  - •Solving linear systems of equations in three variables by elimination
  - •Solving linear systems of equations in three variables by substitution
- •Vector equations of lines
- •Linear subspace
- •Orthogonal complements
- •Orthogonal projection
- ▸The four fundamental spaces
  - •Column space
  - •Nullspace, left nullspace, and row space
- •Van Aubel's theorem
- •Visualizing a projection onto a plane
- •Bretschneider's formula

› Linear algebra

Proof: Cross product distributes over addition

\(\vec{a} \times (\vec{b} + \vec{c}) = (\vec{a} \times \vec{b}) + (\vec{a} \times \vec{c})\)

7942Vector Cross Product Distributes over Addition

ProofWiki