MA 351: Introduction to Linear Algebra and Its Applications
Fall 2024, Midterm One
Instructor: Yip
This test booklet has FIVE QUESTIONS, totaling 100
points for the whole test. You have 75 minutes to do
this test.
Plan your time well. Read the questions
carefully.
This test is
closed book, closed note, with no electronic
devices.
In order to get full credits, you need to give
correct and
simplified answers and explain in a
comprehensible way how you arrive at
them.
As a rule of thumb, you should give explicit and
useful answers.
No points will be given for just writing down some
generically true statements. In other words, your
answers should try to make use of all the information
given in the question.
As a rule of thumb, you should only use those
methods that have been covered in class.
If you use some other methods "for the sake of
convenience", at our discretion, we might not give you
any credit. You have the right to contest. In that
event,
you will be asked to explain your answer using only
what has been covered in class up to the point of
time of this exam.
I have read the above instructions. Signature
Name: Answer Key (Major: )
Question and Score
Question
Score
1.(20 pts)
2.(20 pts)
3.(20 pts)
4.(20 pts)
5.(20 pts)
Total (100 pts)
Consider the equation \( 2x - 3y = 5 \).
Write down the solution of the above equation in
parametric form as \( X = p + \mathrm{Null}(A)
\) using some vector \( p \) and null space of
some matrix \( A \). You should write \(
\mathrm{Null}(A) \) as a span of vector(s). Give
two different examples of \( p \).
For each of the two forms of solution you have
found, plot your solution in the following \( xy
\)-plane. Clearly indicate the vector \( p \)
and \( \mathrm{Null}(A) \).
Figure 1
A hand-annotated xy-coordinate grid with
axes labeled x (horizontal) and y
(vertical), each marked from about -3 to
3. A red dashed line through the origin
with positive slope is labeled Null(A)
in blue. A solid red line parallel to
it, shifted down and to the right,
passes through the points \(
(\frac{5}{2}, 0) \) and \( (0,
-\frac{5}{3}) \); it is labeled in red
as \( 2x - 3y = 5 \) and, below the
graph, as shifted Null(A). A red arrow
from the origin along the positive
x-axis to \( (\frac{5}{2}, 0) \) is
labeled \( p_1 \) with the column vector
\( \begin{pmatrix} \frac{5}{2} \\ 0
\end{pmatrix} \). A second red arrow
from the origin straight down the
negative y-axis to \( (0, -\frac{5}{3})
\) is labeled \( p_2 \) with the column
vector \( \begin{pmatrix} 0 \\
-\frac{5}{3} \end{pmatrix} \). Red
arrowheads along the solid line indicate
the direction of the solution line
through both endpoints.
You are given the following list of polynomials from
\( \mathbb{P}_2 \), the space of polynomials of
degree at most 2, \( \mathbb{P}_2 = \{p(x) = a + bx
+ cx^2 : a, b, c \in \mathbb{R}\} \):
Determine if the above list is linearly independent.
If not, throw away all the redundant vector(s) until
you get a linearly independent list. For each vector
you throw away, write it as a linear combination of
the vectors from the linearly independent list you
have found.
\( p_3, p_4, p_5 \) — redundant, can be written as
lin. comb. of \( p_1 \) and \( p_2 \).
You are given a linearly independent list \(
\mathcal{A} = \{X_1, X_2, X_3\} \) from some vector
space. Consider the new list \( \mathcal{B} = \{Y_1,
Y_2, Y_3\} \) where
Is \( \mathcal{B} \) linearly independent? If so,
write any linear combination of the \( X_i \)'s as a
linear combination of the \( Y_i \)'s. If not,
determine which of the \( Y_i \)'s are redundant.
Consider a region with two main economies: M
(machinery) and S (service). From historical data,
it is found out that:
producing 1 unit of M requires 0.1 unit of M
and, 0.2 unit of S;
producing 1 unit of S requires 0.1 unit of M
and, 0.3 unit of S.
Suppose the total outside demands of M and S are 40
and 50 units, respectively. Determine the actual
units of M and S produced in order to satisfy the
above demands.
Figure 2
A hand-drawn input-output flow diagram.
Two circled nodes are drawn side by
side: node M on the left, labeled in red
with the variable \( x \) beneath the
letter M, and node S on the right,
labeled in red with the variable \( y \)
beneath the letter S. A curved self-loop
arrow above node M is labeled 0.1, and a
curved self-loop arrow above node S is
labeled 0.3. A long curved arrow runs
from S leftward into M, labeled 0.2
above it. Another long curved arrow runs
from M rightward into S, labeled 0.1
below it. An arrow points left out of
node M toward the number 40 written at
the far left, representing the outside
demand for machinery. An arrow points
right out of node S toward the number 50
at the far right, representing the
outside demand for service.
(In class we talked about the property of linear
dependence as the presence of some redundant
vectors. The following is another
equivalent definition in terms of
uniqueness of representations.) Given a list \(
\mathcal{L} = \{u_1, u_2, \ldots u_n\} \) of vectors
from some vector space. Prove the following two
statements about \( \mathcal{L} \).
Suppose you have two different,
unequal sets of numbers \(
\{\alpha_1, \alpha_2, \ldots \alpha_n\} \)
and \( \{\beta_1, \beta_2, \ldots \beta_n\}
\) such that
Suppose \( \mathcal{L} \) is linearly
dependent. Let \( v \) be a vector from \(
\operatorname{Span}\{\mathcal{L}\} \)
written as \( v = \alpha_1 u_1 + \alpha_2
u_2 + \cdots \alpha_n u_n \). Show that
there is another, different,
representation of \( v \) as a linear
combination of the \( u_i \)'s.
How many different representations of \( v
\) can you find? How do they all look like?
Note: parts (a) and (b) above are two separate
statements.
