Question

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.

Answer #1

For each of the following statements, say whether the statement
is true or false.
(a) If S⊆T are sets of vectors, then span(S)⊆span(T).
(b) If S⊆T are sets of vectors, and S is linearly independent,
then so is T.
(c) Every set of vectors is a subset of a basis.
(d) If S is a linearly independent set of vectors, and u is a
vector not in the span of S, then S∪{u} is linearly
independent.
(e) In a finite-dimensional...

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...

(6) Label each of the following statements as True or
False. Provide justification
for your response.
(b) True/False The scalar λ is an eigenvalue of a
square matrix A if and
only if the equation (A − λIn)x = 0 has a nontrivial
solution.
(c) True/False If λ is an eigenvalue of a matrix A, then there is
only
one nonzero vector v with Av = λv.
(d) True/False The eigenspace of an eigenvalue of an n × n matrix...

Determine if the following statements are true or false.
In either case, provide a formal proof using the definitions of
the big-O, big-Omega, and big-Theta notations. For instance, to
formally prove that f (n) ∈ O(g(n)) or f (n) ∉ O(g(n)), we need to
demonstrate the existence of a constant c and a sufficient large n0
such that f (n) ≤ c g(n) for all n ≥ n0, or showing that there are
no such values.
a) [1 mark] 10000n2...

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...

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

Evaluate the following statements as true or false and provide
your reasoning. [5 pts. each]
When MPL is less than APL, APL
will fall.
When MPL equals zero, APL will also equal
zero.

For each of the following statements clearly
indicate whether the statement is true or false. Provide a
brief explanation to justify your answer. Your answer
must begin with "True" or
"False" followed by your explanation. Note
that your explanation cannot just be a restatement of the question
statement.
a) A security with a beta of 0.8 can have a standard deviation
of return that is greater than the market portfolio’s standard
deviation of return. (Begin your answer with
"True" or...

1. Indicate if each of the following is true or false. If false,
provide a counterexample.
(a) The mean of a sample is always the same as the median of
it.
(b) The mean of a population is the same as that of a
sample.
(c) If a value appears more than half in a sample, then the mode
is equal to the value.
2. (Union and intersection of sets) Let Ω = {1,2,...,6}. Suppose
each element is equally likely,...

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