Linear Algebra Definitions

Linear Algebra Definitions Questions and Answers 2023. linear transformation Correct Answers: (from a vector space V into a vector space W): A rule T that assigns to each vector x in V a unique vector T(x) in W such that (i) T (u+v) = T(u) + T(v) (ii) T(cu) = cT(u) for all u in V and all scalars c standard matrix Correct Answers: for a linear transformation T: R^n - R^m, there exists a unique matrix A such that T(x) = Ax for all x in R^n. This is called the standard matrix. A = [T(e1) ... T (en)] one-to-one Correct Answers: A mapping T: R^n - R^m is one-to-one if each b in R^m is the image of at most one x in R^n. onto Correct Answers: A mapping T:R^n - R^m is onto if each b in R^m is the image of at least one x in R^n. subspace Correct Answers: A subspace of a vector space V is a subset H of V that has three properties: a. The zero vector of V is in H b. H is closed under vector addition. That is, for each u and v in H, the sum u+v is in H. c. H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector cu is in H. adjugate Correct Answers: The matrix adj A formed from a square matrix A by replacing the (i,j) entry of A by the (i,j) cofactor, for all i and j, and then transposing the matrix elementary matrix Correct Answers: An invertible matrix that results by performing one elementary row operation on an identity matrix transpose Correct Answers: The transpose of a matrix A (of dimension m x n) is the n x m matrix whose columns are the corresponding rows of A kernel Correct Answers: the kernel of a linear transformation T: V - Wis the set of x in V such that T(x) = 0 null space Correct Answers: The null space of an m x n matrix A is the set of all solutions to Ax = 0 column space Correct Answers: The column space of an m x n matrix A is the set of all linear combinations of the columns of A. Col A = span {a1, a2,... an} where ai represents a column of A