<p>M 340 L Unique number 53280 QUIZ # 4</p><p>TRUE FALSE</p><p>1) The columns of P CB are linearly independent. TRUE</p><p>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</p><p>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. </p><p>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. </p><p>6) A matrix with orthonormal columns is orthogonal. FALSE Doubtless if an orthogonal matrix has orthonormal columns but should also be square.</p><p>7) If x is orthogonal to every vector in a subspace W, then x is in W. TRUE</p><p>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.</p><p>9) If |v + u 2 =  v 2 + || u ||2 , then u and v are orthogonal. TRUE</p><p>10) Not every linearly independent set in Rn is an orthogonal set. TRUE</p><p>11) Not every orthogonal set in Rn is linearly independent. (*)</p><p>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.</p><p>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.</p><p>15) For an mxn matrix A, vectors in the null space of A are orthogonal to vectors in the row space of A. TRUE</p><p>(*) 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.</p>