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
whenever i  j, then S is an orthonormal set.
Such a set is orthogonal, but it is not orthoNORMal.
FALSE
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.