M340L Unique number 53280
Fourth Quiz
Solution
The fourth quiz will take place on Monday, April 24.
Will be in the quiz: Chapter 4 , Sections 4-7, Chapter 5, Sections 5-1, 5-2, 5-3, 5-4.
Chapter 6, 6-1, and 6-2.
To get ready for this quiz, you may study in your book:
Section 4.7, page 270, Exercises # 11, 12.
Section 5.1, page 303, Exercises # 21, 22,
and 23 – 27.
Section 5.2 page 312, Exercises # 20, 21, 22. Section 5.3 page 320, Exercises # 21, 22,
23 - 28.
Section 5.4 page 328, Exercises # 19, 20, 21, 22, 23, 24.
(2 square matrices A and B are similar if there exists an invertible matrix P such that
B = P-1 A P . In other words A and B are similar if they represent the same linear
transformation in different bases.)
Supplementary exercises 1, 2, 3, page 365.
Section 6.1, page 377, Exercises # 19, and 20. Section 6.2, page 387, Exercises # 23, 24,
29, 30, 32.
The questions will be chosen from these exercises.
Try to get the answers by yourselves before you look at the solutions below.
First Practice quiz.
TRUE FALSE
The columns of the change of coordinates matrix P CB are
B-coordinate vectors of the vectors in C.
It is just the contrary: the columns are the vectors in B written in the
C-basis. (Remember the formula [x] E = B[x] B)
FALSE
2) The columns of P CB are linearly independent.
These columns are the vectors of a basis and are linearly independent.
TRUE
3) A matrix A is not invertible if and only if 0 is an eigenvalue of A.
There exists v  0 such that Av = 0 v = 0.
TRUE
4) The eigenvalues of a matrix are on its main diagonal.
It would be true only for triangular matrices.
FALSE
5) An eigenspace of A is a null space of a certain matrix.
E =Ker (A- I)
TRUE
6) If +5 is a factor of the characteristic polynomial of A, then 5 is an
eigenvalue of A.
It is –5.
FALSE
7) A row replacement operation on A does not change the eigenvalues.
If B is an echelon form of A, in general det (A- I)  det(B- I)
FALSE
8) If A is a diagonalizable nxn matrix then A has n distinct eigenvalues.
However if A has n distinct eigenvalues it is diagonalizable.
FALSE
9) For any scalar c, we have:  c v = c v.
It is true only when c is positive.
FALSE
10) Not every linearly independent set in Rn is an orthogonal set.
TRUE
2nd practice quiz
1)Not every orthogonal set in Rn is linearly independent.
FALSE
1) The orthogonal projection of y onto v is the same as the orthogonal
projection of y onto c v where c  0.
TRUE
2) If U and V are orthogonal matrices (U -1= U tand V -1= V t ) then UV is
orthogonal.
TRUE
t
t t
-1 -1
-1
(UV) = V U = V U = (U V)
4) A square orthogonal matrix is invertible.
TRUE
5) For an mxn matrix A, vectors in the null space of A are orthogonal to
vectors in the row space of A.
TRUE
A x = 0 which means that x is in Ker A, also means that x is orthogonal to
the rows of A.
6) For a square matrix A, vectors in Col A are orthogonal to vectors in
Nul A.
FALSE
7) If Rn has a basis of eigenvectors of A, then A is diagonalizable.
TRUE
8) If AP = PD , with D diagonal, then the non zero columns of P must be
eigenvectors of A.
TRUE
9) If v1 and v2 are linearly independent vectors, then they correspond to
distinct eigenvalues.
Think of the case when  () = 2 and dim (E ) = 2.
10) If A2 is the zero matrix, then the only eigenvalue of A is 0.
A x = s x , A2 x = s A x = s2 x = 0 with x  0 brings s = 0.
FALSE
TRUE