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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 › Basic transformation matrices

    Composition of transformations

    In this section, you'll learn how transformation matrices can be composed. Before attempting this section, you should understand how matrices can be used to transform a point. You should also be very comfortable multiplying matrices.

    YouTube videos

    • 2441Matrix Transformations : reflections and rotations
      Maths Videos - by jayates