Term
The row space of A is the same as the column space of A^{T}. 

Definition


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


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


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


Term
The row space of A^{T} is the same as the column space of A. 

Definition


Term
If A and B are row equivalent, then their row spaces are the same. 

Definition


Term
The columns of the changeofcoordinates matrix P C<B are Bcoordinate vectors of the vectors in C. 

Definition
false, it's the Ccoordinate vectors of the vectors in the basis B 


Term
If V = R^{n} and C is the standard basis for V, then P C<B is the same as the changeofcoordinates matrix P_{B} 

Definition


Term
The columns of P C<B are linearly independent. 

Definition


Term
If V = R^{2}, B = {b_{1}, b_{2}}, and C = {c_{1}, c_{2}}, then row reduction of [c_{1} c_{2} b_{1} b_{2}] 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


Term
A number c is an eigenvalue of A if and only if the equation (A  cI)x = 0 has a nontrivial solution. 

Definition


Term
Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy. 

Definition


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 v_{1} and v_{2} are linearly independent eigenvectors, then they correspond to distinct eigenvalues 

Definition
false, two linearly independent vectors can correspond to the same eigenvalue 


Term
A steadystate vector for a stochastic matrix is actually an eigenvector. 

Definition


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

Definition


Term
If λ+5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A. 

Definition


Term
If A is a square matrix with columns a_{1}, a_{2}, a_{3}, then detA equals the volume of the parallelepiped determined by a_{1}, a_{2}, a_{3}. 

Definition
false, the detA equals the volume of the parallelepiped determined by a_{1}, a_{2}, a_{3}, not detA. 


Term

Definition


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


Term
A row replacement operation on A does not change the eigenvalues. 

Definition


Term
A is diagonalizable if A = PDP^{1 }for some matrix D and some invertible matrix P. 

Definition


Term
If R^{n} has a basis of eigenvectors of A, then A is diagonalizable. 

Definition


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


Term
If A is invertible, then A is diagonalizable. 

Definition
false, a matrix can be invertible and not diagonalizable 


Term

Definition


Term
For any scalar c, u·(cv) = c(u·v) 

Definition


Term
If the distance from u to v equals the distance from u to v, then u and v are orthogonal. 

Definition


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 v_{1},,,,,v_{p} span a subspace W and if x is orthogonal to each v_{j} for j = 1....p, then x is in W^{[image]}. 

Definition


Term

Definition


Term
For any scalar c, cv = cv 

Definition
false, cv = c v, not cv 


Term
If x is orthogonal to every vector in a subspace W, then x is in W^{[image]}. 

Definition


Term
If u^{2} + v^{2 }= u + v^{2}, then u and v are orthogonal. 

Definition


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

