HW2: Math 405. Due: beginning of class, Wednesday, 7 September 2016 1. Write out the identity matrix in R4. 2. Find α, β, γ such that 2 sin α− cos β + 3 tan γ = 3 4 sin α + 2 cos β − 2 tan γ = 2 6 sin α− 3 cos β + tan γ = 9 for 0 ≤ α < 2π, 0 ≤ β < 2π, 0 ≤ γ < π. 3. Find coefficients a,b,c,d so that the curve shown is given by y = ax3 + bx2 + cx + d, and passes through the points: (0, 10), (1, 7), (3,−11), (4,−14). Note: This problem requires row-reducing a 4x5 matrix. You can do it by hand, but for this problem, maybe you want to use some software. If you don’t know anything else, you can use Wolfram Alpha at http://www.wolframalpha.com/. An example: If you want to use Gauss-Jordan elimination on the matrix A =  1 2 34 5 6 7 8 9   , then you would type in: “RowReduce[{{1, 2, 3},{4, 5, 6},{7, 8, 9}}]” with all the capitals, square and curly brackets, and commas (but not quotation marks). Wolfram would spit back out at you:  1 0 −10 1 2 0 0 0   . 4. Consider the following system of linear equations. 3x1 + 10x2 + x3 = 6 −x1 + 2x2 + 3x3 = 4 x1 + 6x2 + 2x3 = 5 1 (a) Find the homogeneous solution. (b) Find the particular solution. (c) What is the full solution. 5. Consider a set V = {v ∈ R3|v1 + v2 = 0; v1 + 2v2 + v3 = 10}. (a) Is V a vector space? Why or why not? (b) If V is a vector space, determine its dimension and find a basis. 6. Consider a set V = {v ∈ R3|v1 + v2 = 0; v1 + 2v2 + v3 = 0}. (a) Is V a vector space? Why or why not? (b) If V is a vector space, determine its dimension and find a basis. 7. Is the set of vectors V = {v1,v2,v3} with v1 =  12 2   ; v2 = [−1 0 3]T ; v3 = [0 −4 −2]T linearly independent? Why or why not? If dependent, show the dependence. 8. Is the set of vectors V = {v1,v2,v3} with v1 =   12 −1   ; v2 = [−1 0 −1]T ; v3 = [0 −4 4]T linearly independent? Why or why not? If dependent, show the dependence. 9. Given the matrix A =   9 0 1 0 0 0 1 0 1 1 1 1 0 0 1 0   . (a) What is the rank? (b) What is the dimension of the row space? One obvious choice of basis for the row space is BA = { 9 0 1 0   ,   0 0 1 0   ,   1 1 1 1   } . Another is Brr = { 1 0 0 0   ,   0 1 0 1   ,   0 0 1 0   } . Show that you can describe the basis vectors of BA in terms of the basis vectors of Brr. (c) What is the dimension of the column space? Find a basis for the column space. (d) What is the dimension of the null space? Find a basis for the null space. 2 10. Given the same matrix A =   9 0 1 0 0 0 1 0 1 1 1 1 0 0 1 0   . Is the vector v = [ [2 1 −1 −1 ]T a member of its row space? Show why or why not. 11. Consider the set, P3, of polynomials in x ∈ R of degree 3. So, P3 = {a0 + a1x + a2x2 + a3x3 for a0 ∈ R,a1 ∈ R,a2 ∈ R,a3 ∈ R}. (a) Is P3 a vector space? Why or why not? (b) If so, what is it’s dimension? (c) If so, what is a basis? 12. Consider the ODE d2y(x) dx2 + b2y(x) = 0 for fi.

1. HW2: Math 405. Due: beginning of class, Wednesday, 7
September 2016
1. Write out the identity matrix in R4.
2. Find α, β, γ such that
2 sin α− cos β + 3 tan γ = 3
4 sin α + 2 cos β − 2 tan γ = 2
6 sin α− 3 cos β + tan γ = 9
for 0 ≤ α < 2π, 0 ≤ β < 2π, 0 ≤ γ < π.
3. Find coefficients a,b,c,d so that the curve shown is given by
y = ax3 + bx2 + cx + d,
and passes through the points: (0, 10), (1, 7), (3,−11), (4,−14).
Note: This problem requires row-reducing a 4x5 matrix. You
can do it by hand,
but for this problem, maybe you want to use some software. If
you don’t know
anything else, you can use Wolfram Alpha at
http://www.wolframalpha.com/.
An example: If you want to use Gauss-Jordan elimination on the
matrix
A =

2. 7 8 9
then you would type in: “RowReduce[{{1, 2, 3},{4, 5, 6},{7, 8,
9}}]” with all
the capitals, square and curly brackets, and commas (but not
quotation marks).
Wolfram would spit back out at you:
0 0 0
4. Consider the following system of linear equations.
3x1 + 10x2 + x3 = 6
−x1 + 2x2 + 3x3 = 4
x1 + 6x2 + 2x3 = 5
1
(a) Find the homogeneous solution.
(b) Find the particular solution.
(c) What is the full solution.

3. 5. Consider a set V = {v ∈ R3|v1 + v2 = 0; v1 + 2v2 + v3 =
10}.
(a) Is V a vector space? Why or why not?
(b) If V is a vector space, determine its dimension and find a
basis.
6. Consider a set V = {v ∈ R3|v1 + v2 = 0; v1 + 2v2 + v3 = 0}.
(a) Is V a vector space? Why or why not?
(b) If V is a vector space, determine its dimension and find a
basis.
7. Is the set of vectors V = {v1,v2,v3} with
v1 =
2
linearly independent? Why or why not? If dependent, show the
dependence.
8. Is the set of vectors V = {v1,v2,v3} with
v1 =

4. −1
linearly independent? Why or why not? If dependent, show the
dependence.
9. Given the matrix
A =
9 0 1 0
0 0 1 0
1 1 1 1
0 0 1 0
(a) What is the rank?
(b) What is the dimension of the row space? One obvious choice
of basis for the row
space is BA =
9
0
1
0

5. 0
0
1
0
1
1
1
1
}
. Another is Brr =
1
0
0
0

6. 0
1
0
1
0
0
1
0
}
.
Show that you can describe the basis vectors of BA in terms of
the basis vectors
of Brr.
(c) What is the dimension of the column space? Find a basis for
the column space.
(d) What is the dimension of the null space? Find a basis for the
null space.
2

7. 10. Given the same matrix
A =
9 0 1 0
0 0 1 0
1 1 1 1
0 0 1 0
Is the vector v =
[
[2 1 −1 −1
]T
a member of its row space? Show why or why
not.
11. Consider the set, P3, of polynomials in x ∈ R of degree 3.
So,
P3 = {a0 + a1x + a2x2 + a3x3 for a0 ∈ R,a1 ∈ R,a2 ∈ R,a3 ∈
R}.
(a) Is P3 a vector space? Why or why not?
(b) If so, what is it’s dimension?
(c) If so, what is a basis?

8. 12. Consider the ODE
d2y(x)
dx2
+ b2y(x) = 0 for fixed b ∈ R. (1)
(a) Show that (1) has solutions y(x) = p cos(bx) + q sin(bx), ∀
(p,q) ∈ R.
(b) Let V be the set of all solutions to (1). Show that V is a
vector space.
(c) What is a basis for V ? What is the dimension of V ?
3
Copyright © Amina Khalifa El-Ashmawy 2014
Identifying of an Unknown Substance
Amina Khalifa El-Ashmawy, Ph.D.
Collin College
Department of Chemistry
Introduction:
Elements and compounds are pure substances. A compound
consists of two or more
elements that are chemically bonded in definite proportion.
Based on the constituent
elements, we can fairly accurately determine the type of

9. compound it is.
Each compound has its own set of physical and chemical
properties, which depend largely
on the type of compound. Ionic substances tend to have a high
melting point, can be
soluble in water and tend to be fairly dense. Physical properties
of covalent compounds
range widely based on the size and polarity of the molecule.
Density is a measure of how tightly packed the particles are in a
substance. On the
macroscopic level, density is the mass of a sample in a given
amount of space (volume).
Density = mass/volume
The units for density depend on the state of the substance.
Usually, for solids and liquids,
the units are g/mL. Since volume fluctuates with temperature,
density is a temperature
dependent property. For reference, water’s density is about 1.0
g/mL. When two
substances are mixed, the less dense substance will float and the
more dense substance will
sink.
If a substance is solid at room temperature, its melting point
(mp) is higher than room
temperature, and it can be measured fairly accurately. If a
substance is a liquid at room
temperature, its mp is lower than room temperature. Both its
freezing point and boiling
point can be measured. Lastly, if a substance is gas at room

10. temperature, its boiling point
is lower than room temperature.
Solubility depends on the type of substance the solute and
solvent are. If the solute and
solvent are similar in their molecular characteristics (i.e. one
ionic and one polar; both are
polar; or both are nonpolar), the solute will dissolve in the
solvent. If the two are dissimilar
in their molecular characteristics (i.e. one is either ionic or
polar and the other is
nonpolar), the solute will be insoluble in the solvent.
Physical properties allow us to identify an unknown substance,
which is the purpose of this
lab.
Properties of Water:
Water is an interesting compound in many ways. It is the only
common compound whose
density as a solid is less than its density as a liquid. Actually,
the maximum density of
water, 1.000 g/mL, is at 4oC. Table 1 lists the density of water
at different temperatures.
Copyright © Amina Khalifa El-Ashmawy 2014
Table 1. Density of Water at Various Temperatures

11. Temperature
(oC)
Density
(g/mL)
0 0.9998
4 1.0000
10 0.9997
20 0.9982
30 0.9957
40 0.9922
50 0.9881
60 0.9832
70 0.9778
80 0.9718
90 0.9653
100 0.9584
Water forms a meniscus when placed in glass containers. A
meniscus is the curve at the
surface of a liquid that is close to the container. When
measuring volume, one should be
eye level with the meniscus and read the volume corresponding
to the bottom of the
meniscus for concave meniscus and at the top of the meniscus
for a convex meniscus.
Another interesting property of water is the wide range of
temperature for which it exists
as a liquid at standard pressure. Specifically, water’s melting or
freezing point is at 0oC
while its boiling point at standard pressure is 100oC.

12. The last property we will address is water’s function as a
solvent. In fact, water is
considered the universal solvent because of its ubiquity on
Earth. For solid substances
that are not soluble in water (i.e. predominately nonpolar
substances), water displacement
can be used to determine the volume of those solids.
The Problem:
Each pair of students will have one solid and one liquid
unknown to identify based on their
physical properties. See Table 2.
Measuring Melting Point and Boiling Point:
To measure melting point, place your unknown in a test tube,
suspend a thermometer just
at the surface of the solid and insert the test tube into a beaker
containing a hot water bath.
Make sure the test tube is not pointed towards anyone.
To measure boiling point, place your unknown in a test tube
with one or two boiling chips,
suspend a thermometer just above the surface of the liquid, and
heat gently to boiling.
Make sure the test tube is not pointed towards anyone.
Copyright © Amina Khalifa El-Ashmawy 2014
Table 2. Physical Properties of Miscellaneous Substances

13. Substance
Density
(g/mL)
Melting Point
(oC)
Boiling Point
(oC)
Solubility
Water Ethanol
Borax 1.73 75 dec 200 Sl-s I
Calcium carbonate 2.93 Dec 825 dec I I
Calcium nitrate 1.82 43 dec S S
Cyclohexane 0.78 6.5 81 I S
p-Dichlorobenzene 1.46 53 174 I S
Diphenylmethane 1.00 27 265 I S
Ethanol 0.79 -112 78 S S
Heptane 0.68 -91 98 I Sl-s
Hexane 0.66 -94 69 I S
Iodomethane 2.28 -66 42 Sl-s S
Lauric acid 0.88 48 225 I S
Methanol 0.79 -98 65 S S
Naphthalene 1.15 80 218 I Sl-s
2-Propanol 0.79 -86 83 S S
Propanone 0.79 -95 56 S S
Stearic acid 0.85 70 291 I Sl-s
Thymol 0.97 52 232 Sl-s S
p-Toluidine 0.97 45 200 Sl-s S
Trichloromethane 1.49 -63.5 61 I S
Zinc nitrate 2.06 36 dec 105 S S
Note: Symbols used in this table are S = soluble, Sl-s = slightly

14. soluble, I = insoluble, dec = decomposes. Values are rounded.
Prelab Questions:
x What is a physical property?
x What properties does one use to identify a liquid substance?
Solid?
x How do you measure the volume of a solid unknown substance
that is either a
powder or granular crystals?
x Is there a necessary order for measuring the physical
properties?
x When observing solubility in the two solvents, what else will
you be observing?
x What is the best way to measure melting point? Boiling point?
(You should research
this before coming to lab.)
x If an unknown has properties that are similar to more than one
possible compound,
what are some ways to distinguish the unknown?
Critical Data/Discussion to Include in Your Lab Report:
x Data table showing melting point, boiling point, density and
solubility, including
unknown number and room temperature
x Discussion section that includes the answer to all questions
posed above, the
results, and the identity of your unknown substance along with
its unknown
number (if your unknown could be more than one substance
based on Table 2,

15. discuss the possible identities)
x Possible sources of error
x Works cited