Linear Algebra and Equations Questionnaire

here is the textbook,

https://math.berkeley.edu/~yonah/files/Linear%20Al…

cover sec 1.1-1.8 and sec 2.1, the exam

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Math 18, Midterm Exam, Summer 2021
Duration: 2:30-4:00
Show all your work; no credit will be given for unsupported answers based on the material
presented in the lectures.
1. What relation must 𝑎, 𝑏, and 𝑐 satisfy so that the following system is consistent for all values
of ℎ and 𝑘?
𝑥 𝑎𝑦 ℎ
𝑏𝑥 𝑐𝑦 𝑘
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2. Let 𝐴 be an 𝑚 𝑛 matrix and let 𝑏 be a vector in 𝑅 .
a. If the system 𝐴𝑥 𝑏 has a unique solution, must the system 𝐴𝑥
b. If the system 𝐴𝑥 0 has a unique solution, must the system 𝐴𝑥
0 have a unique solution?
𝑏 be consistent?
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3. Let 𝑀 be an 𝑚 𝑛 matrix. Supposed 𝑀𝑣
𝑢 and 𝑀𝑣
𝑀𝑥 𝑟𝑢
𝑡𝑢 consistent for all scalars 𝑟 and 𝑡?
𝑢 . Is the system
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4. Let 𝐴 be a 7 9 augmented matrix of a system of linear equations and let 𝐵 be the coefficient
matrix of the system.
a. If the 9 column of 𝐴 is a pivot column, is the system consistent?
b. If the 7th row of an echelon form of 𝐵 has a leading entry, is the system consistent?
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5. Let 𝑣 , 𝑣 , 𝑣 , 𝑣 be vectors in 𝑅 , where 𝑣 , 𝑣 , 𝑣 are linearly independent and 𝑣 is not in the
𝑆𝑝𝑎𝑛 𝑣 , 𝑣 , 𝑣 . Must 𝑣 , 𝑣 , 𝑣 , 𝑣 be linearly independent?
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6. Construct a non-homogeneous system with 3 equations and 3 unknowns, such that all the
1
entries of its coefficient matrix are different from each other and 𝑥
2 is a solution of the
3
system.
il
Exercise 2.1
Consider the system of equations
2X1 + 1×2 + x3 = 1
1×1 + 0x2 + 3×3 = 2
-3X1 + 2X2-7X3 = 3
d. Solve it by hand,
Convert this system of equations into a matrix equation of the form Cx
and record your solution in your document.
a. Enter the matrix C and the column vector d into MATLAB, and use the command
>> X
Cid
to check your solution.
b. We would expect to get the column vector d in MATLAB if we ran the command C*X, right?
In other words, C*x-d should be zero. Enter this expression into MATLAB:
>> C*X-d
Include the input and output.
Exercise 2.2
Consider the system of equations
-10X1 + 5×2 = 0
6X1 – 3×2 = 0
As you did in the previous exercise, enter the corresponding matrix C and column vector d into
MATLAB. Then type in
>> X =
Cid
Note the strange output. Include it in your write-up. Now go ahead and solve this system by
hand. How many free variables do you have in your solution? Based on your answer, can you
explain why you got the error message when trying to use the command x C\d?
=
Exercise 2.3
while
Consider the homogeneous system of equations
X1 – 3×2 + 2×3 = 0
-2X1 + 6×2 – 4×3 = 0
4×1 – 12X2 + 8×3 = 0
By using the rref command, write down the general solution to this system of equations. How
many free variables are required?
Wassily Leontief (1906-1999) was a Russian-born American economist who, aside from developing
highly sophisticated economic theories, enjoyed trout fishing, ballet, and fine wines. He won the 1971
Nobel Memorial Prize in Economics for his work in creating mathematical models to describe various
economic phenomena. In the remainder of this lab we will look at a very simple special case of his
work called a closed exchange model. There are two basic assumptions:
Everyone exclusively buys from and sells to the central pool (i.e., there is no outside supply or
demand).
Everything produced is consumed.
With these assumptions and some data about how the goods are consumed, we can compute exactly
what price each good should have for everyone in the community to survive.
To see how this works, let’s suppose there’s a small country town with only five residents: a farmer, a
tailor, a carpenter, a coal miner, and Slacker Bob. The farmer produces food; the tailor, clothes; the
carpenter, housing; the coal miner, energy; and Slacker Bob makes moonshine, half of which he drinks
himself. The following table lists what fraction of each good our five residents consume:
Food
Clothes
Housing Energy Alcohol
Farmer
0.25
0.15
0.25
0.18
0.20
Tailor
0.17
0.28
0.18
0.17
0.10
0.22
0.19
0.22
0.22
0.10
Carpenter
Miner
0.20
0.15
0.20
0.28
0.15
Slacker Bob
0.16
0.23
0.15
0.15
0.45
So for example, the carpenter consumes 22% of all food, 19% of all clothes, 22% of all housing, 22%
of all energy, and 10% of all the moonshine.
Exercise 2.4
The columns in this table all add up to 1. Explain why.
LINEAR ALGEBRA
AND ITS APPLICATIONS
FIFTH EDITION
DAVID C. LAY
STEVEN R. LAY
JUDI J. MCDONALD
Suppose Ax = 0 has a unique solution; this solution has to be x = 0.
Let u and u be solutions to Ax = b. Then Alu – U) = Au – Av = b – b = 0,
this is, u – u is a solution of Ax 0. Since x = O is the unique solution of
Ax = 0, then it has to be u – v = 0, this is, u = v. Hence Ax b also has a
unique solution (if one exists).
=
Let z be the unique solution of Ax b, and let y be a nonzero solution for
Ax 0. Then, by linearity, A(z + y) = Az + Ay = b + 0 = b. Hence, z + y is
another solution for Ax = b. Contradiction.
=
If Uk is
It means
not be represented
f. Yes,
not in the span
of lV., U2, U3), then there is
no solution to the equation X, U + X Vzt X3 V3 =0
V4 Can
as the Umear Combination of (vi, Us, U3)
It U, Us, Us are Limearly independent, then Xivit XV 27 X 3V3=0
has only the trurial solution Combrinity, the two notion it
Vu is
span of (vi, V, V3)
Xivit X₂ V2 + X₃ U ₃ + X4 V4
will
have
only
the trural solution
Hence IV, V, Vs , V4) must be Umearly to
not
also
you need to explain why:
consider wether x4 could be o
at independent.