# Midterm Exam Solved

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1. (10 points) Determine whether each of the following statements is true or false. No justification is required.
a) If matrices A and B commute, then matrices A T and BT must commute as well.

(b) If matrix A is symmetric and matrix S is orthogonal, then matrix S −1AS must be symmetric.

(c) There exists a subspace V of R5 such that dim(V ) = dim(V ⊥).

(d) If two n × n matrices A and B are similar, then the equation det(A) = det(B) must hold.

(e) If a real matrix A has only the eigenvalues 1 and −1, then A must be orthogonal

2.  (10 points) Find the least-squares solution ~x ∗ of the system A~x = ~b, where
A =

1 1
1 0
0 1

 and ~b =

3
3
3

3. (10 points) Find the orthogonal projection of ~x =1 0 0 0T
onto the subspace of R4spanned by

1
1
1
1

,

1
1
−1
−1

,

1
−1
−1
1

4. (10 points) Consider an n × m matrix A. Find dim(im(A)) + dim(ker(AT )) in terms of m and n

5. (10 points) Find the QR-factorization of the following matrix: A =4 25 00 0 −23 −25 0

6. (10 points) Consider the linear space P1 with the following inner product: hf , gi =Z 10f (t)g(t)d t

(a) Find an orthonormal basis of this space

(b) Find the orthogonal projection onto P1 of f (t) = t2.

7. (10 points) Calculate the determinant of the following matrix:
A =

1 1 1 1
1 1 4 4
1 −1 2 −2
1 −1 8 −8

8. Given a set of n functions f1, . . . , fn, the Wronskian W(f1, . . . , fn) is given by:
W(f1, . . . , fn) = det



f1 f2 · · · fn
f

1
f

2
· · · f

n
f
′′
1
f
′′
2
· · · f
′′
n
.
.
.
.
.
.
.
.
.
.
.
.
f
(n−1)
1
f
(n−1)
2
· · · f
(n−1)
n


The Wronskian can be used to determine whether a set of differentiable functions is linearly independent
on a given interval. If the Wronskian is nonzero at some point in an interval, then the associated functions
are linearly independent.
(a) (5 points) Calculate the Wronskian of {x, cos x, sin x}.

(b) (5 points) Calculate the Wronskian of {2x 2 + 3, x2, 1}.

9. Consider a 2 × 2 matrix of the form A =

a b
c dwhere a, b, c, d are positive numbers such that a + c =
b + d = 1. Such a matrix is called a regular transition matrix.
(a) (5 points) Verify that 
b
c

and 
1
−1

are eigenvectors of A, and find the associated eigenvalues.

(b) (5 points) Using part (a), find a closed formula for the components of the following dynamical system
with initial value ~x0 = ~e1:
~x(t + 1) = 
0.5 0.25
0.5 0.75
~x(t)

10. (10 points) Consider an invertible n×n matrix A. Give five different equivalent statements about A. Trivial
statements such as “3A is invertible” or “AT is invertible” are not acceptable.

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