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MA212 Final Exam - Linear Algebra II
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MCQs on Linear Algebra.

MA212 Final Exam - Linear Algebra II
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25 Questions

1. Which of the following expresses the complex number
in polar coordinates?
2. What is the Riesz representation theorem?
3. Let
be a linear transformation. Suppose the matrix for
relative to a basis
for
is
. Suppose
is the transition matrix from another basis
to
. Then the matrix for
with respect to
is
4. What are the eigenvalues of
?
5. Which of the following is a basis of
?
6. According to the Cauchy-Schwarz inequality, on any inner product space which of the following must be true?
7. Let
be such that
for all
written in the standard basis. Then,
is a
8. Which of the following is an example of a normal operator?
9. Calculate the eigenvalues of
.
10. Let
be subspaces, then
, if and only if two conditions hold. One is that
. What is the other condition?
11. What are the proper subspaces of
?
12. Suppose
is a nonsingular
matrix. Then which of the following best describes its four fundamental subspaces?
13. If
is the union of the coordinate axes, then
is NOT a real vector space because
14. A Hilbert space is
15. An
matrix
is called a Markov matrix if the following is satisfied:
16. Which of the following fields has the Archimedean property?
17. Suppose
and
are two distinct eigenvalues of a real symmetric matrix
. Then what are their corresponding eigenvectors?
18. Let
and let
be defined by
. Then what is the matrix,
, for
using the standard basis?
19. Fill in the blanks. Consider the polynomial
. Counted with multiplicity, this polynomial has ________ roots in
and ________ roots in
.
20. Which of the following is NOT a vector space associated with the
matrix
?
21. Which of the following properties should a norm on a normed linear space satisfy?
22. Given
, the adjoint of
is defined to be the operator
, such that which is true?
23. If
and
are symmetric matrices, then the product
is also symmetric only when
and
are
24. Consider the following linear transformation
:
→
where
rotates each vector 90 degrees counterclockwise about the
-axis and then rotates 45 degrees counterclockwise about the
-axis. Find the matrix representation of
in the standard basis.
25. Let
. Determine the Jordan normal form for
.

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