Practice problems for midterm 2, Math 225 R1
Fall 2012
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Name :
1
1. Compute the determinants using a cofactor expansion
3 0 4 (a) 2 3 2 0 5 −1 2 −4 3
3 1 2
1 4 −1
2. Answer the following questions. Give reasons for your answer.
(a) What is the determinant of an elementary row replacement matrix?
(b) What is the determinant of an elementary scaling matrix with k on
diagonal?
2
3. Let A =
a b
and k be a scalar. Find a formula that relates det kA
c d
and det A.
4. Each equation illustrates a property of determinants. State the property.
0 5 −2 1 −3 6 1 −3 6 = − 0 5 −2 4 −1 8 4 −1 8 1 3 −4 1 3 −4 2 0 −3 = 0 −6 5 5 −4 7 5 −4 7 3
5. Find the determinants by row reduction to echelon form.
1 5 −6 (a) −1 −4 4 −2 −7 9 (b) 1 3 0 2
−2 −5 7 4
3 5 2 1
1 −1 2 −3
6. Find the
a
(a) d
5g
a b c
determinants followings, where d e f
g h i
b c e f 5h 5i a b c
(b) g h i
d e f
4
=7
7. Find a formula for det (rA), where A is an n × n matrix.
8. Use Cramer’s rule to compute the solutions of the systems.
(
5x1 + 7x2 = 3
(a)
2x1 + 4x2 = 1
(
3x1 − 2x2 = 7
(b)
−5x1 + 6x2 = −5


2x1 + x2 = 7
(c) −3x1 + x3 = −8


x2 + 2x3 = −3
5
9. Determine the values of the parameter s for which the system has a
unique solution, and describe the solution.
(
6sx1 + 4x2 = 5
(a)
9x1 + 2sx2 = −2
(b)
(
sx1 − 2sx2 = −1
3x1 + 6sx2 = 4
6
10. Compute the adjugate of the given matrix, and then give the inverse of
the matrix.


0 −2 −1
(a)  3 0 0 
−1 1 1


3 5 4
(b) 1 0 1
2 1 1
7
11. Let V be the first quadrant in the xy-plane; that is let V = {(x, y) : x ≥
0, y ≥ 0}.
(a) If u and v are in V , is u + v in V ?
(b) Find a specific vector u and specific scalar c such that cu is not in
V . (This shows that V is not a vector space.)
12. Let H be the set of points inside and on the unit circle in the xy-plane.
That is, let H = {(x, y) : x2 + y 2 ≤ 1}. Find a specific example - two
vectors or a vector and a scalar - to show that H is not a subspace of
R2 .
8
13. Determine if the given set is a subspace of Pn for an appropriate value
of n.
(a) All polynomials in Pn such that p(0) = 0.
(b) All polynomials of the form p(t) = a + t2 , where a ∈ R.
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

−2t
14. Let H be the set of all vectors of the form  5t . Find a vector v ∈ R3
3t
such that H = Span{v}. Explain why this shows H is a subspace of R3 .

2b + 3c
15. Let W be the set of all vectors of the form  −b  where b, c are
2c
arbitrary real numbers. Find vectors u, v such that W = Span{u, v}.
Explain why this shows W is a subspace of R3 .

10
16. The set of all continuous real-valued functions defined on a closed interval
[a, b] in R is denoted by C[a, b]. This set is a subspace of the vector space
of all real-valued functions defined on [a, b].
(a) What facts about continuous functions should be proved in order to
demonstrate that C[a, b] is indeed a subspace as claimed?
(b) Show that {f ∈ C[a, b] : f (a) = f (b)} is a subspace of C[a, b].
17. Find an explicit description of Nul A, by listing vectors that span the
null space.
1 2 4 0
(a) A =
0 1 3 −2


1 −4 0 2 0
(b) A = 0 0 1 −5 0
0 0 0 0 2
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18. Either use an appropriate theorem to show that the given set W , is a
vector space, or find a specific example to the contrary.

 

 a
(a)  b  : a + b + c = 2


c

 
p





 
q

(b) 
r  : p − 3q = 4s, 2p = s + 5r





s
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19. Find A such that the given set is Col A.



2s
+
t







r
−
s
+
2t


(a) 
 : r, s, t ∈ R
3r
+
s





2r − s − t



b
−
c







2b
+
3d

(b) 
b + 3c − 3d : b, c, d ∈ R





c+d
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20. Find a basis for the set of vectors in R3 in the place x − 3y + 2z = 0.
[Hint : Think of the equation as a ”system” of homogeneous equations.]
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21. Assume
toB.
 that A is row equivalent

−2 4 −2 −4
1
A =  2 −6 −3 1 , B = 0
−3 8 2 −3
0
Find bases
 for Nul A and Col A.
0 6 5
2 5 3.
0 0 0
 
 

7
1
4
22. Let v1 = −3, v2 =  9 , v3 = 11, and also let H = Span{v1 , v2 , v3 }.
6
−2
7
It can be verified that 4v1 + 5v2 − 3v3 = 0. Use this information to find
a basis for H.

15
23. Mark each statement True or False.
(a) A single vector by itself is linearly dependent.
(b) If H = Span{b1 , b2 , · · · , bp }, then {b1 , · · · bp } is a basis for H.
(c) The columns of an invertible n × n matrix form a basis for Rn .
(d) In some cases, the linear dependence relations among the columns
of a matrix can be affected by certain elementary row operations on
the matrix.
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24. Consider the polynomials p1 (t) = 1+t2 and p2 (t) = 1−t2 . Is {p1 (t), p2 (t)}
a linearly independent set in P3 ?
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