Linear Algebra Definitions Flashcards

Linear Algebra Definitions

• Linear Transformation: A function T from vector space V to W that satisfies two main properties—additivity (T(u + v) = T(u) + T(v)) and homogeneity (T(cu) = cT(u)).

• Standard Matrix: For a linear transformation T from R^n to R^m, there is a unique matrix A that represents T such that T(x) = Ax for all x in R^n, where A = [T(e1)...T(en)].

• One-to-One Mapping: A mapping T from R^n to R^m is classified as one-to-one if each b in R^m corresponds to at most one vector x in R^n.

• Onto Mapping: A mapping T from R^n to R^m is considered onto if every vector b in R^m is the image of at least one vector x in R^n.

• Subspace: A subset H of a vector space V qualifies as a subspace if it includes the zero vector, is closed under vector addition, and is closed under scalar multiplication.

• Adjugate Matrix: Formed from a square matrix A by replacing each entry with its cofactor and transposing the result.

• Elementary Matrix: An invertible matrix created by performing a single elementary row operation on an identity matrix.

• Transpose of a Matrix: The resulting matrix from switching the rows and columns of matrix A, transforming it into dimensions n x m from m x n.

• Kernel: The kernel of a linear transformation T: V -> W consists of all vectors x in V for which T(x) equals the zero vector.

• Null Space: The collection of all solutions to the equation Ax = 0 for an m x n matrix A.

• Column Space: Represents all possible linear combinations of the columns of an m x n matrix A, denoted as Col A.

• Row Space: The set of all linear combinations of the rows of matrix A, equivalent to the column space of A transposed.

• Linear Independence: A set of vectors {v1...vp} is linearly independent if the only solution to c1v1 + ... + cpvp = 0 is when all coefficients c1, ..., cp are zero.

• Basis: A linearly independent indexed set B = {v1...vp} that spans a subspace H, indicating H = span{v1...vp}.

• Spanning Set: A collection {v1...vp} in subspace H that satisfies the condition H = span{v1...vp}.

• Dimension: Denotes the size of a basis for vector space V, represented as dim V; the dimension of the zero vector space is 0.

• Isomorphism: A linear mapping that establishes a one-to-one correspondence between two vector spaces.

• Rank: Defined as the dimension of the column space of matrix A, indicating the maximum number of linearly independent columns.

• Change-of-Coordinates Matrix: A matrix that transforms coordinate vectors from one basis B to another basis C.

• Eigenvalue: A scalar λ for which the equation Ax = λx has solutions for some nonzero vector x.

• Eigenvector: A nonzero vector x satisfying the equation Ax = λx for a given scalar λ.

• Characteristic Polynomial: For matrix A, expressed as det(A - λI), it provides the eigenvalues through its roots.

• Similar Matrices: Two matrices A and B are similar if there exists an invertible matrix P such that A = PBP^-1.

• Diagonalizable Matrix: A matrix that can be decomposed into the form PDP^-1, where D is diagonal and P is invertible.

• Inner Product: The scalar product of two vectors u and v, denoted as u^Tv or u.v, also known as the dot product.

• Orthogonal Matrix: A square matrix U where U^-1 equals U^T; such matrices feature orthonormal columns.

• Orthonormal Set: A set of vectors that are both orthogonal and unit vectors, meaning the dot product between any two different vectors in the set equals zero.

• Orthogonal Projection: The orthogonal projection of a vector y onto another vector u is calculated using the formula (y.u)/(u.u).