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Differential Equations and Linear Algebra 2250
Midterm Exam 2
Version 1, 21 Mar 2014
Instructions: This in-class exam is designed to be completed in 30 minutes. No calculators, notes, tables or
books. No answer check is expected. Details count 3/4, answers count 1/4.
1. (The 3 Possibilities with Symbols)
Let a, b and c denote constants and consider the system of equations
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b—3
2b—4
i—b
b—i
a
a
0
a
2
3
—i
1
=
z
b
—
2
b
2
b
—b
+b
2
b
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(a) [40%] Determine a and b such that the system has a unique solution.
(b) [30%] Explain why a
=
0 and b
=
b
0 implies no solution. Ignore any other possible no solution cases.
4n:
(c) [30%] Explain why a
=
0 implies infinitely many solutions. Ignore any other possible infinitely
many solution cases.
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2250 Midterm Exam 2 S2014 [2lMar]
Name.
2. (Vector Spaces) Do all parts. Details not required for (a-(d).
()( )()
(a) [10%] or fhlse: The span of the vectors
is a subspace of
of
,
,
0
dimension 2.
t
,
(b) [10%] True or A matrix equationA b has a solution if and only if the matrix A has an
VM
inverse.
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(c) [10%] rue or false: Relation x + y = 2x + z defines a subspace in 7?.
1\
(d) [10%] rue or false: The span of er, e, sinh(x) is a subspace of the vector space of functions
continuous on —oo <x <
(e) [20%] State two linear algebra theorems that apply to conclude that the set S = span(1, x, x2) is a
subspace of the vector space of all polynomials. Do not present details, but please state the theorems
fully.
(f) [40%] Find a basis of vectors for the subspace of 7? given by the system of restriction equations
.
311
+
+
2x
+
211
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2250 Midterm Exam 2 S2014 [2lMar]
Name.
/
3.
(Independence and Dependence)
Do all parts.
(a) [10%] State an independence test for 3 vectors in R.
. Write the hypothesis and conclusion, not just
5
the name of the test.
(b) [10%] State fully an independence test for 3 functions, each is which is a linear combination of Euler
solution atoms, e.g., e, cosh(x), sin(x) + x. No details are expected, but please state the test fully.
(c) [40%] Let v
, v
1
, v
2
, v
3
4 denote the rows of the matrix
A—
—
0
0
0
0
—2
1
2
1
—6 0 0
2 1 0
510
301
Decide if the four rows
, v
2
i
,v
3
4 are independent and display the details of the chosen independence
test.
(d) [40%] Extract from the list below a largest set of independent vectors.
,
j=
0
0
0
0
0
0
2
2
—2
2
,=
0
1
1
—1
1
,=
0
1
1
1
3
0
3
3
—1
5
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0
0
0
2
2
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2250 Midterm Exam 2 52014 [2lMar]
Name.
4. (Determinants) Do all parts.
(a) [10%] True o The toolkit operations of swap, combination and multiply apply to determinants
7
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and leave the value of the determinant unchanged.
(b) [10%] True or For 3 x 3 matrices A and B, ]A+ B = 1A + B.
(c) [20%] Assume given 3 x 3 matrices A, B. Suppose A
, B
2
3 are elementary
B =E
2
1 and E
2
3
, E
1
matrices representing respectively a combination, a multiply by —3 and a swap. Assume det(B)
9.
Let C = 3A. Find all possible values of det(C).
(d) [30%] Suppose matrix C is 8 x 8, nonzero but C
2 = 0. Let B = I C and A = I + C. Explain why
B is the inverse of A.
(e) [30%] Let symbols a, b, c denote constants. Define matrix
r 8.
—r X2
—
A—
—
llOa
1 0 0 0
abOl
1 c12
0. Then the adjugate [adjoint] formula for the inverse is defined. Find the value of the
entry in row 2, column 3 of A’.
Assume ]A
4(3
C= (-XO=i
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c
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1
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C
-
3
-
—
b’.-N)
0
Cl 2
i.c.
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2250 Midterm Exam 2 S2014 [2lMar]
Name.
5. (Linear Differential Equations) Do all parts.
(a) [20%] Solve for the general solution of 12y” -I- hg’ + 2
y
=
0.
5)2
(b) [40%] The characteristic equation is (4r
4 2
)(r 2r +
r
= 0. Find the general solution y of the
linear homogeneous constant-coefficient differential equation.
(c) [20%] A fourth order linear homogeneous differential equation with constant coefficients has a par
ticular solution 2e + sin(x) + xe. Find the roots of the characteristic equation.
(d) [20%] Mark with X the functions which can possibly be be a solution of a linear homogeneous
differential equation with constant coefficients. Test your choices against this theorem:
—
—
The general solution of a linear homogeneous nth order differential equation with constant
coefficients is a linear combination of Euler solution atoms.
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)
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±2
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