Linear algebra - Study guide

1. Linear equations
   1. Linear equations in one variable
   2. Linear equations in two variables
   3. Simultaneous linear equations
   4. Solving linear equations
   5. Solutions of a linear equation
   6. Graphing linear equations
   7. Applications of linear equations
2. Matrices
   1. Matrix operations
      1. Matrix addition and subtraction
      2. Matrix multiplication and division
      3. Scalar multiplication and division
      4. Transpose of a matrix
      5. Trace of a matrix
      6. Determinant of a matrix
      7. Inverse of a matrix
      8. Adjoint of a matrix
      9. Cofactor of a matrix
      10. Minor of a matrix
   2. Matrix properties
      1. Commutative property
      2. Associative property
      3. Distributive property
      4. Identity matrix
      5. Zero matrix
      6. Diagonal matrix
      7. Triangular matrix
      8. Symmetric matrix
      9. Skew-symmetric matrix
      10. Orthogonal matrix
   3. Matrix methods
      1. Gaussian elimination method
      2. Gauss-Jordan method
      3. Cramer's rule method
   4. Matrix applications
      1. Systems of linear equations
      2. Linear transformations
      3. Encryption and decryption
3. Vectors
   1. Vector operations
      1. Vector addition and subtraction
      2. Scalar multiplication and division
      3. Dot product and cross product
      4. Magnitude and direction of a vector
      5. Unit vector and normal vector
   2. Vector properties
      1. Commutative property
      2. Associative property
      3. Distributive property
      4. Zero vector
      5. Parallel vectors
      6. Perpendicular vectors
   3. Vector methods
      1. Component form of a vector
      2. Position vector and displacement vector
      3. Projection of a vector
   4. Vector applications
      1. Geometry and physics problems
4. Vector spaces
   1. Vector space axioms
   2. Subspaces and span of a vector space
   3. Linear independence and dependence of vectors
   4. Basis and dimension of a vector space
   5. Row space, column space, and null space of a matrix
   6. Rank and nullity of a matrix
5. Inner product spaces
   1. Inner product and norm of a vector
   2. Orthogonal and orthonormal vectors
   3. Gram-Schmidt process and QR decomposition
   4. Orthogonal complement and projection
6. Linear transformations
   1. Definition and properties of linear transformations
   2. Kernel and image of a linear transformation
   3. Matrix representation of a linear transformation
7. Eigenvalues and eigenvectors
8. Special matrices