Glossary
- Page ID
- 72278
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Words (or words that have the same definition) | The definition is case sensitive | (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages] | (Optional) Caption for Image | (Optional) External or Internal Link | (Optional) Source for Definition |
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(Eg. "Genetic, Hereditary, DNA ...") | (Eg. "Relating to genes or heredity") | The infamous double helix | https://bio.libretexts.org/ | CC-BY-SA; Delmar Larsen |
Word(s) |
Definition |
Image | Caption | Link | Source |
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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) |
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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 |