M 340 L Unique Number 53280 QUIZ # 4

M 340 L Unique number 53280 QUIZ # 4

TRUE FALSE

1) The columns of P CB are linearly independent. TRUE

2) A number c is an eigenvalue of A if and only if the equation (A – cI) x = 0 has a non trivial solution. TRUE

3) A is diagonalizable if and only if A has n eigenvalues, counting multiplicities. FALSE The characteristic polynomial of any nxn matrix (diagonalizable or not) is of degree n and has n real or complex roots, counting multiplicities.

4) If a 3x3 matrix A has 2 eigenvalues and if each eigenspace is one-dimensional then A is diagonalizable. FALSE If A is 3x3, its characteristic polynomial is of degree 3 and has 3 roots counting with multiplicities. If there are only 2 eigenvalues, one of them is of multiplicity 2, and the associated eigenspace should be 2-dimensional for A to be diagonalizable.

6) A matrix with orthonormal columns is orthogonal. FALSE Doubtless if an orthogonal matrix has orthonormal columns but should also be square.

7) If x is orthogonal to every vector in a subspace W, then x is in W. TRUE

8) If A is diagonalizable then A has n distinct eigenvalues. FALSE It is true that if A has n distinct eigenvalues it is diagonalizable but the converse is not true.

9) If |v + u 2 =  v 2 + || u ||2 , then u and v are orthogonal. TRUE

10) Not every linearly independent set in Rn is an orthogonal set. TRUE

11) Not every orthogonal set in Rn is linearly independent. (*)

12) An orthogonal matrix is invertible. TRUE The students who have answered TRUE in 6) and FALSE in 12) although not correct are consistent, I’ve given them something for this consistency.

13) If a set S = {u1, …. , up} has the property that ui .uj = 0 FALSE whenever i  j, then S is an orthonormal set. Such a set is orthogonal, but it is not orthoNORMal. 14) If v1 and v2 are linearly independent vectors, then they correspond to distinct eigenvalues. FALSE If 2 vectors correspond to distinct eigenvalues they are linearly independent but the converse is not true. Think of ()= 2 and dim E=2.

15) For an mxn matrix A, vectors in the null space of A are orthogonal to vectors in the row space of A. TRUE

(*) I have accepted both False and True. Actually the 0 vector is orthogonal to any vector, so that one might consider an orthogonal set containing the 0 vector, and in that case the orthogonal set would be linearly dependent. However when one considers an orthogonal set it is always an orthogonal set of non 0 vectors which then is linearly independent. The question being quibbling, I have decided to accept both answers.