Dr. Mark V. Sapir
List of concepts and definitions
- Linear equation
- Solution of a linear equation
- System of linear equations
- Solution of a system of linear equations
- A consistent system of equations
- General solution of a system of linear equations
- Operations on systems of linear equations
- Augmented matrix of a system of lin. equations
- Row operations on matrices
- Row-echelon form
- Leading 1 in a row echelon matrix
- Reduced row-echelon form
- Leading unknown
- Free unknown
- Back-substitution
- Gauss-Jordan elimination procedure
- Homogenious system of lin. equations
- Size of a matrix
- Entry of a matrix
- Row-vector
- Column-vector
- Square matrix
- Scalar
- Equality of matrices
- Sum of matrices
- Multiplying a matrix by a scalar
- Product of a row-vecror and a column-vector
- Product of matrices
-
Matrix of coefficients of a system of equations.
- Zero matrix
- Identity matrix
- Transpose of a matrix
- Trace of a matrix
- Invertible matrix
- Inverse of a matrix
- Elementary matrix
- The inverse of an elementary matrix
- Symmetric matrices
- Diagonal matrices
- Upper- and lowertriangular matrices
- Skew-symmetric matrices
- A transposition
- Determinant of a square matrix
- Sign of a permutation
- An inversion in a
permutation.
- Even and odd permutations.
- Cofactor of entry aij.
- Minor of entry aij.
- Cofactor expansion
along a row (column)
- The adjoint of a matrix
- Cramer's rule
- Algebraic system
- n-space
- n-vector
- Coordinates of an n-vector
- Vector space
- Triangle
- Dot product
(Euclidean inner product) of n-vectors
- Dot product (inner product) of
functions
- Properties of dot
product
- Euclidean vector space
- Norm of a vector
- Distance between two vectors
- Orthogonal vectors
- Linear
transformation from Rn to Rm
- Linear operator on Rn
- Standard matrix of a linear transformation
- The product of linear transformations
- The sum of linear transformations
- Injective maps
- Surjective maps
- General linear
transformations
- General linear operator
- Null transformation
- Identity operator
- Subspace of a vector space.
- Kernel of a linear transformation
- Range of a linear transformation
- Linear combination of vectors
- Vector space spanned by a set of vectors
- The null space of a
matrix
- Column space of a matrix
- Linearly independent sets of elements in a vector space.
- Linearly dependent sets of elements in a vector space.
- The Wronskian of a set of functions
- A basis of a vector space
- The dimension
of a vector space
- Coordinates of a vector in a given basis
- Rank of a set of vectors
- Core of a set of vectors
- Orthogonal complement of a subspace
- Orthogonal basis
- Gram-Schmidt procedure
- Projection of a vector onto a subspace
- Normal component of a vector
- Distance from a vector to a set in a Euclidean vector space
- A least squares
solution of a system of linear equations
- A transition matrix from
one basis to another
- The matrix of a linear operator in a given basis
- Similar matrices
- Eigenvector and eigenvalue of a linear operator
- Eigenvector and eigenvalue of a matrix
- The characteristic polynomial of a matrix