# Shared Flashcard Set

## Details

True/False Questions Matrices Test 3
True and false questions
50
Mathematics
11/15/2010

Term
 The row space of A is the same as the column space of AT.
Definition
 true
Term
 If B is any echelon form of A, and if B has 3 nonzero rows, then the first 3 rows of A form a basis for Row A.
Definition
 false, row operations do not preserve dependence relations so it would wrong to say that because B has 3 linearly independent rows, that A has 3 linearly independent rows also. The first 3 rows of B form a basis for Row A.
Term
 The dimensions of the row space and the column space of A are the same, even if A is not square.
Definition
 true
Term
 The sum of the dimensions of the row space and the null space of A equals the number of rows in A.
Definition
 false, the sum equals the number of columns in A.
Term
 On a computer, row operations can change the apparent rank of a matrix.
Definition
 true
Term
 If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.
Definition
 false, the pivot columns of A form a basis for the column space of A.
Term
 Row operations preserve the linear dependence relations among the rows of A.
Definition
 false, row operations do not preserve the linear dependence relations among the rows of A.
Term
 The dimension of the null space of A is the number of columns of A that are not pivot columns.
Definition
 true
Term
 The row space of AT is the same as the column space of A.
Definition
 true
Term
 If A and B are row equivalent, then their row spaces are the same.
Definition
 true
Term
 The columns of the change-of-coordinates matrix P C<--B are B-coordinate vectors of the vectors in C.
Definition
 false, it's the C-coordinate vectors of the vectors in the basis B
Term
 If V = Rn and C is the standard basis for V, then P C<--B is the same as the change-of-coordinates matrix PB
Definition
 true
Term
 The columns of P C<--B are linearly independent.
Definition
 true
Term
 If V = R2, B = {b1, b2}, and C = {c1, c2}, then row reduction of [c1 c2 b1 b2] to [I P] produces a matrix P that satisfies [X]B = P[X]C for all x in V.
Definition
 False, it satisfies [X]C = P[X]B
Term
 If Ax = λx for some vector x, then λ is an eigenvalue of A.
Definition
 false, the vector has to be nonzero
Term
 A matrix A is not invertible if and only if 0 is an eigenvalue of A.
Definition
 true
Term
 A number c is an eigenvalue of A if and only if the equation (A - cI)x = 0 has a nontrivial solution.
Definition
 true
Term
 Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy.
Definition
 true
Term
 To find the eigenvalues of A, reduce A to echelon form
Definition
 false, to find eigenvalues of a matrix you can reduce the matrix to triangular form. Row reducing to reduced echelon form will help you find the eigenvectors
Term
 If Ax = λx for some scalar λ, then x is an eigenvector of A.
Definition
 false, the vector x has to be nonzero
Term
 If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues
Definition
 false, two linearly independent vectors can correspond to the same eigenvalue
Term
 A steady-state vector for a stochastic matrix is actually an eigenvector.
Definition
 true
Term
 The eigenvalues of a matrix are on its main diagonal
Definition
 false, the matrix has to be triangular
Term
 An eigenspace of A is a null space of a certain matrix
Definition
 true, this certain matrix is A - λI
Term
 The determinant of A is the product of the diagonal entries of A.
Definition
 false, this is only true if A is triangular
Term
 An elementary row operation on A does not change the determinant.
Definition
 false, an interchange of two rows changes the determinant which is an elementary row op.
Term
 (detA)(detB) = detAB
Definition
 true
Term
 If λ+5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.
Definition
 false, -5 is
Term
 If A is a square matrix with columns a1, a2, a3, then detA equals the volume of the parallelepiped determined by a1, a2, a3.
Definition
 false, the |detA| equals the volume of the parallelepiped determined by a1, a2, a3, not detA.
Term
 detAT= (-1) detA
Definition
 false, detAT= detA
Term
 The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A
Definition
 true
Term
 A row replacement operation on A does not change the eigenvalues.
Definition
 false, it does
Term
 A is diagonalizable if A = PDP-1 for some matrix D and some invertible matrix P.
Definition
 false, D has to diagonal
Term
 If Rn has a basis of eigenvectors of A, then A is diagonalizable.
Definition
 true
Term
 A is diagonalizable if and only if A has n eigenvalues, counting multiplicities.
Definition
 false, A has to have n distinct eigenvalues
Term
 If A is diagonalizable, then A is invertible.
Definition
 false, A can be diagonalizable and not invertible
Term
 A is diagonalizable if A has n eigenvectors.
Definition
 false, the eigenvectors have to linearly independent
Term
 If A is diagonalizable, then A has n distict eigenvalues.
Definition
 false, this is the converse of Theorem 6. A matrix can be diagonalizable and not have n distinct eigenvalues.
Term
 If AP=PD with D diagonal, then the nonzero columns of P must be eigenvectors of A.
Definition
 true
Term
 If A is invertible, then A is diagonalizable.
Definition
 false, a matrix can be invertible and not diagonalizable
Term
 v·v = ||v||2
Definition
 true
Term
 For any scalar c, u·(cv) = c(u·v)
Definition
 true
Term
 If the distance from u to v equals the distance from u to -v, then u and v are orthogonal.
Definition
 true
Term
 For a square matrix A, vectors in Col A are orthogonal to vectors in Nul A
Definition
 false, vectors in Col A are not orthogonal to vectors in Nul A. Counterexample: [ 1 1     0 0 ]
Term
 If vectors v1,,,,,vp span a subspace W and if x is orthogonal to each vj for j = 1....p, then x is in W[image].
Definition
 true
Term
 u·v - v·u = 0
Definition
 true
Term
 For any scalar c, ||cv|| = c||v||
Definition
 false, ||cv|| = |c| ||v||, not c||v||
Term
 If x is orthogonal to every vector in a subspace W, then x is in W[image].
Definition
 true
Term
 If ||u||2 + ||v||2 = ||u + v||2, then u and v are orthogonal.
Definition
 true
Term
 For an m x n matrix A, vectors in the null space of A are orthogonal to vectors in the row space of A.
Definition
 true
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