Linear Algebra Definitions Questions and Answers 2023.
linear transformation Correct Answers: (from a vector space V into a vector space W): A rule T
... [Show More] 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
row space Correct Answers: The set Row A of all linear combinations of the vectors formed from the rows of A; also denoted by Col A ^T
linear independence Correct Answers: (for a set of vectors) An indexed set {v1...vp} with the property that there exist weights c1...cp all equal to zero such that c1v1 +...+cpvp = 0
basis Correct Answers: An indexed set B = {v1...vp} in V such that
(i) B is a linearly independent set and [Show Less]