Proof Index
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Linear algebra
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Orthogonal projection
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Linear subspace
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Visualizing a projection onto a plane
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Orthogonal complements
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The four fundamental spaces
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Column space
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Nullspace, left nullspace, and row space
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Van Aubel's theorem
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Adding and subtracting matrices
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Adding vectors in polar form
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Applications of linear algebra
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Applications of vectors
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Area of a parallelogram using determinant
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Area of a triangle using determinant
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Are the vectors parallel, orthogonal, or neither?
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Basic transformation matrices
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Transforming a point
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Composition of transformations
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Basis
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Basis for a null space
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Basis for a span
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Cramer's rule
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Solving linear systems in two variables using Cramer's rule
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Solving linear systems in three variables using Cramer's rule
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Definition of \(\mathbb{R}^n\)
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Determinants
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Determinant using row echelon form
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Determinant using cofactors
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Determinant using row operations
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Determinant of a \(4{\times}4\) matrix using row operations
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Determinant of a \(3{\times}3\) matrix
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Determinant of a \(3{\times}3\) matrix using expansion by minors
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Determinant of a \(3{\times}3\) matrix using the diagonal method
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Determinant of a \(2{\times}2\) matrix
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Area of a triangle using determinants
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Intuition: Determinants
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Diagonalization
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Eigenvalues and eigenvectors
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Power method
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Shifted inverse power method
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Finding the eigenvalues and eigenvectors of \(2{\times}2\) matrices
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Finding the eigenvalues and eigenvectors of \(3{\times}3\) matrices
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Finding the eigenvalues and eigenvectors of \(4{\times}4\) matrices
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Elementary matrices
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Applications of the dot product
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Gaussian elimination
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Geometric proofs using vectors
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Proof: Law of cosines using the dot product
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Proof: Rhombus diagonals are perpendicular bisectors (linear algebra)
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Proof: Varignon's theorem (linear algebra)
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Proof: Thales's theorem using the dot product
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Gershgorin circle theorem
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History
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Length of a vector
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Linear combinations
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Linear transformations
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Linear transformations on \(\mathbb{R}^2\)
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Magnitude and direction form of vectors
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Lesson
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Practice
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Matrix inverses
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Matrix multiplication
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Intuition: Matrix multiplication
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Multiplying a matrix by a scalar
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Practice
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Multiplying a vector by a scalar
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Orthogonal vectors
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Parallel vectors
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Proof: Cauchy–Schwarz inequality
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Proof: Cross product distributes over addition
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Proof: Dot product formula
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Proof: Scalar multiplication is associative
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Proof: Vector addition is commutative and associative
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Proof the cross product is distributive
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Algebraic proof the cross product is distributive
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Geometric proof the cross product is distributive
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Properties of matrix multiplication
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Properties of scalar multiplication
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Section formula
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Solving linear systems of equations
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Solving linear systems of equations using matrix inverses
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Solving linear systems of equations in two variables using matrix inverses
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Solving linear systems of equations in three variables using Gaussian elimination
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Consistency of a system of linear equations
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Gaussâ€“Jordan elimination
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Solving linear systems of equations in two variables using Gauss–Jordan elimination
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Solving matrix equations
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Transpose and its properties
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Unit vectors
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Finding the unit vector
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Vector magnitude
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Vector projection
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Proof: Vector projection formula
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Vectors
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3D vectors
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3D vector operations
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Adding and subtracting vectors
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Adding vectors
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Subtracting vectors
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Components of vectors
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Lesson
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Practice
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Equivalent vectors
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Properties of vector addition
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Properties of vector addition and scalar multiplication
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What is an orthogonal set?
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What is a vector space?
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Zero matrix
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William Rowan Hamilton
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Solving linear systems of equations in three variables
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Solving linear systems of equations in three variables (word problems)
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Solving linear systems of equations in three variables by elimination
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Solving linear systems of equations in three variables by substitution
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Vector equations of lines
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Linear algebra
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Eigenvalues and eigenvectors
Finding the eigenvalues and eigenvectors of \(4{\times}4\) matrices
YouTube videos
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Ex: Find the Eigenvalues of a 4x4 Matrix
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