Question

Determine if the following statement is true or false.
**Explain your reasoning.**

If A if an m x n matrix with linearly independent columns, then m is greater than or equal to n.

Answer #1

For each of the following statement, determine it is “always
true” or “always false”. Explain your answer.
(a) For any matrix A, the matrices A and −A have the same row
space. (b) For any matrix A, the matrices A and −A have the same
null space.
(c) For any matrix A, if A has nullity 0, then A is
nonsingular.

Is the following statement true, false or uncertain? Explain
your reasoning. Feel free to draw a diagram with social welfare
curves and utilities possibilities frontier.
“One conclusion from welfare economics is that society will
never prefer an inefficient allocation of resources to an efficient
allocation of resources.”

(part a)
Is the following statement TRUE or FALSE? Explain your
reasoning.
"If the average total costs for producing a good rise with the
quantity produced, there would be diseconomies of
scale."
(part b)
Think of General Motors (car manufacturer), UPS (delivery
service), and Nike (sports apparel manufacturer). Which of these
three companies do you think would be most likely to have
economies of scope? Explain your
answer.

Mark the following as true or false, as the case may be. If a
statement is true, then prove it. If a statement is false, then
provide a counter-example.
a) A set containing a single vector is linearly independent
b) The set of vectors {v, kv} is linearly dependent for every
scalar k
c) every linearly dependent set contains the zero vector
d) The functions f1 and f2 are linearly
dependent is there is a real number x, so that...

5. Determine whether the following statements are TRUE or FALSE.
If the statement is TRUE, then explain your reasoning. If the
statement is FALSE, then provide a counter-example. a) The
amplitude of f(x)=−2cos(X- π/2) is -2 b) The period of
g(x)=3tan(π/4 – 3x/4) is 4π/3.
. c) If limx→a f (x) does not
exist, and limx→a g(x) does not exist, then limx→a (f (x) + g(x))
does not exist. Hint: Perhaps consider the case where f and g are
piece-wise...

True or False: An n X n matrix A has n linearly
independent eigenvectors. Justify your answer
(explain)
These questions follow the exercises in Chapter 4 of the book,
Scientific Computing: An Introduction, by Michael Heath. They study
some of the properties of eigenvalues, eigenvectors, and some ways
to compute them.

True or False:
An n X n matrix A has n linearly independent eigenvectors.
Justify your answer (explain)
These questions follow
the exercises in Chapter 4 of the book, Scientic Computing: An
Introduction, by Michael Heath. They study some of the properties
of eigenvalues, eigenvectors, and some ways to compute them.

7. Answer the following questions true or false and provide an
explanation. • If you think the statement is true, refer to a
definition or theorem. • If false, give a counter-example to show
that the statement is not true for all cases.
(a) Let A be a 3 × 4 matrix. If A has a pivot on every row then
the equation Ax = b has a unique solution for all b in R^3 .
(b) If the augmented...

3. For each of the following statements, either provide a short
proof that it is true (or appeal to the deﬁnition) or provide a
counterexample showing that it is false.
(e) Any set containing the zero vector is linearly
dependent.
(f) Subsets of linearly dependent sets are linearly
dependent.
(g) Subsets of linearly independent sets are linearly
independent.
(h) The rank of a matrix is equal to the number of its nonzero
columns.

Are the following statements true or false? Please explain why
it's true or false as well.
For a normally distributed random variable, P(X > µ) = .5
Independent variables may be linearly related

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