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Mathematics LibreTexts

Glossary

  • Page ID
    72278
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    Glossary Entries

    Word(s)

    Definition

    Image Caption Link Source
    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 Iby 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        
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