LINEAR ALGEBRA FINAL EXAM
QUESTIONS WITH CORRECT ANSWERS
Algebraic multiplicity - Answer-the multiplicity of an eigenvalue as a root of the
characteristic equation
basic variable - Answer-a variable in a linear system that corresponds to a pivot column
in the coefficient matrix
basis - Answer-an indexed set B = {v1,...,vP} in V such that (i) B is a linearly
independent set and (ii) the subspace spanned by B coincides with H, that is, H =
Span{v1,...,vP}
(maximal linearly independent set)
best approximation - Answer-the closest point in a given subspace to a given vector
characteristic equation - Answer-det(A-λI) = 0
codomain (of a transformation T) - Answer-the set R^m that contains the range of T. In
general, if T maps a vector space V into a vector space W, then W is called the
codomain of T
column space - Answer-the set Col A of all linear combinations of the columns of A. If A
= {a1...aN} then Col A = Span {a1,...,aN}
consistent linear system - Answer-a linear system with at least one solution
diagonalizable - Answer-a matrix that can be written in factored form as PDP⁻¹, where
D is a diagonal matrix and P is an invertible matrix
diagonal matrix - Answer-a square matrix whose entries not on the main diagonal are all
zero
dimension of a subspace S - Answer-the number of vectors in a basis for S
domain (of a transformation T) - Answer-the set of all vectors x for which T(x) is defined
eigenspace - Answer-the set of all solutions of Ax = λx, where λ is an eigenvalue of A.
Consists of the zero vector and all eigenvectors corresponding to λ
eigenvalue - Answer-a scalar λ such that the equation Ax = λx has a solution for some
nonzero vector x
eigenvector - Answer-a nonzero vector x such that Ax = λx for some scalar λ
QUESTIONS WITH CORRECT ANSWERS
Algebraic multiplicity - Answer-the multiplicity of an eigenvalue as a root of the
characteristic equation
basic variable - Answer-a variable in a linear system that corresponds to a pivot column
in the coefficient matrix
basis - Answer-an indexed set B = {v1,...,vP} in V such that (i) B is a linearly
independent set and (ii) the subspace spanned by B coincides with H, that is, H =
Span{v1,...,vP}
(maximal linearly independent set)
best approximation - Answer-the closest point in a given subspace to a given vector
characteristic equation - Answer-det(A-λI) = 0
codomain (of a transformation T) - Answer-the set R^m that contains the range of T. In
general, if T maps a vector space V into a vector space W, then W is called the
codomain of T
column space - Answer-the set Col A of all linear combinations of the columns of A. If A
= {a1...aN} then Col A = Span {a1,...,aN}
consistent linear system - Answer-a linear system with at least one solution
diagonalizable - Answer-a matrix that can be written in factored form as PDP⁻¹, where
D is a diagonal matrix and P is an invertible matrix
diagonal matrix - Answer-a square matrix whose entries not on the main diagonal are all
zero
dimension of a subspace S - Answer-the number of vectors in a basis for S
domain (of a transformation T) - Answer-the set of all vectors x for which T(x) is defined
eigenspace - Answer-the set of all solutions of Ax = λx, where λ is an eigenvalue of A.
Consists of the zero vector and all eigenvectors corresponding to λ
eigenvalue - Answer-a scalar λ such that the equation Ax = λx has a solution for some
nonzero vector x
eigenvector - Answer-a nonzero vector x such that Ax = λx for some scalar λ
