Augmented Matrix |
A coefficient matrix and constant matrix that represents a linear system |
|
|
|
|
Basis of a Vector Space |
A set of vectors, S, is a basis of a vector space, V, if span(S) = V and the vectors in S are linearly independent |
|
|
|
|
Characteristic Polynomial |
Det(cIn-A) where A is an n x n matrix. |
|
|
|
|
Characteristic Equation |
Det(cIn-A)=0 |
|
|
|
|
Column Space of a Matrix |
The vector space spanned by the column vectors of a matrix |
|
|
|
|
Determinant |
A function, or procedure, that assigns an real number to a square matrix |
|
|
|
|
Dimension of a Vector Space |
The number of vectors in a basis |
|
|
|
|
Dot Product |
The scalar product for the vector space Rn |
|
|
|
|
Elementary Matrix |
A n x n matrix E that can be obtained from In by a single row operation |
|
|
|
|
Eigenvalue |
A scalar, c, satisfying AX=cX where A is a square matrix and X is a non-zero vector |
|
|
|
|
Eigenvector |
The vector X in the equation AX=cX |
|
|
|
|
Gaussian Elimination |
A procedure that reduces an augmented matrix to reduced echelon form |
|
|
|
|
Gauss-Jordon Elimination |
A procedure that reduces an augmented matrix to row-reduced echelon form |
|
|
|
|
Homogeneous System |
A system of linear equations for which the constant matrix in the augmented system has only zero entries |
|
|
|
|
Identity Matrix |
A square matrix with ones along the main diagonal and zeros elsewhere |
|
|
|
|
Injective (One-to-One) |
A linear transformation from vector space V to vector space W such that if u,v are vectors in V and T(u)=T(v) then u = v |
|
|
|
|
Inverse Matrix |
A square matrix B is the inverse of square matrix A if AB=BA=In |
|
|
|
|
Inner Product |
A map from VxV to a field F satisfying four specific axioms. |
|
|
|
|
Isomorphism |
A linear transformation that is both one-to-one and onto |
|
|
|
|
Kernel of a Linear Transformation, T, from V to W |
The set of vectors in V such that T(V)= 0, where 0 is the zero vector |
|
|
|
|
Linear Independence |
A set of vectors S in a vector space V is linearly independent if no vector is in the span of a subset of any of the other vectors in S |
|
|
|
|
Linear Transformation |
A function from a vector space V to a vector space W such that, for scalars a, b in V and vectors u, v in V,
T(au+bv)=aT(u)+bT(v)
|
|
|
|
|
LU Factorization of a Matrix M |
Might not always exist but, when possible, M can be written as the product of a lower triangular matrix and an upper triangular matrix |
|
|
|
|
Matrix |
A rectangular array of numbers |
|
|
|
|
Nullity of T |
The dimension of the kernel of a linear transformation T |
|
|
|
|
Orthogonal |
Vectors whose inner product is zero |
|
|
|
|
Orthonormal Basis |
A basis of orthogonal unit vectors |
|
|
|
|
Parameter |
In a system with infinite solutions, used to represent the unrestrained variable(s). Can be any number |
|
|
|
|
Pivot Position |
Location of the leading entry in a row echelon form matrix |
|
|
|
|
Pivot Column |
Column containing the pivot position |
|
|
|
|
Rank of Linear Transformation, T |
The dimension of the image of linear transformation T |
|
|
|
|
Row Space |
The vector space spanned by the row vectors of a matrix |
|
|
|
|
Skew Symmetric Matrix |
A matrix that is equal to the negative of its transpose |
|
|
|
|
Span |
The set of all linear combinations of a set of vectors in a vector space V |
|
|
|
|
Subspace |
A subset W of a vector space V that is itself a vector space |
|
|
|
|
Surjection (Onto) |
A linear transformation T from vector space V to vector space W such that if u is a vector of V there exists w in W with T(u)=w |
|
|
|
|
Symmetric Matrix |
A matrix that is equal to its transpose |
|
|
|
|
Transpose |
An m x n matrix found from n x m matrix A by swapping the rows of A with the columns of A |
|
|
|
|
Unit Vector |
A vector with a length of one |
|
|
|
|
Vector in Rn |
An ordered n-tuple with magnitude and direction |
|
|
|
|
Vector |
An element of a vector space |
|
|
|
|