Question

In each case below show that the statement is True or give an example showing that it is False.

(i) If {X, Y } is independent in R n, then {X, Y, X + Y } is independent.

(ii) If {X, Y, Z} is independent in R n, then {Y, Z} is independent.

(iii) If {Y, Z} is dependent in R n, then {X, Y, Z} is dependent.

(iv) If A is a 5 × 8 matrix with rank A = 4, then dim(null A) = 3.

(v) If A and B are similar matrices, then so are AT and BT .

Answer #1

For each statement below, either show that the statement is true
or give an example showing that it is false. Assume throughout that
A and B are square matrices, unless otherwise specified.
(a) If AB = 0 and A ̸= 0, then B = 0.
(b) If x is a vector of unknowns, b is a constant column vector,
and Ax = b has no solution, then Ax = 0 has no solution.
(c) If x is a vector of...

(a) Let the statement,
∀x∈R,∃y∈R G(x,y), be true for predicate G(x,y).
For each of the following statements, decide if the statement is
certainly true, certainly false,or possibly true, and justify your
solution.
1
(i)
G(3,4)
(ii)
∀x∈RG(x,3)
(iii)
∃y G(3,y)
(iv)
∀y¬G(3,y)(v)∃x G(x,4)

True or false; for each of the statements below, state whether
they are true or false. If false, give an explanation or example
that illustrates why it's false.
(a) The matrix A = [1 0] is not invertible.
[1 -2]
(b) Let B be a matrix. The rowspaces row (B), row (REF(B)) and
row (RREF(B)) are all equivalent.
(c) Let C be a 5 x 7 matrix with nullity 3. The rank of C is
2.
(d) Let D...

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...

1. For each statement that is true, give a proof and for each
false statement, give a counterexample
(a) For all natural numbers n, n2
+n + 17 is prime.
(b) p Þ q and ~ p Þ ~ q are NOT logically
equivalent.
(c) For every real number x
³ 1, x2£
x3.
(d) No rational number x satisfies
x^4+ 1/x
-(x+1)^(1/2)=0.
(e) There do not exist irrational numbers
x and y such that...

Exercise 4.11. For each of the following, state whether it is
true or false. If true, prove. If false, provide a
counterexample.
(i) LetX andY besetsfromRn. IfX⊂Y thenX is closed if and only if
Y is closed.
(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex
then eitherX or Y is closed and convex (one or the other).
(iii) LetX beanopensetandY ⊆X. IfY ≠∅,thenY isaconvexset.
(iv) SupposeX isanopensetandY isaconvexset. IfX∩Y ⊂X then
X∪Y...

1. The dependent variable has a score in the case of logistic
regression. TRUE or FALSE and why?
2, If the probability of an even A is 0.2 and that of an event B
is 0.10 then the odds ratio is: a.1 b. 2.25 c. 2.0 d. 0.5
3. In the case of logistic regression, we estimate how much the
natural logarithm of the odds for Y =1 changes for a unit change in
X. TRUE or FALSE
4. In...

Exercise 4.11. For each of the following, state whether it is
true or false. If true, prove. If false, provide a
counterexample.
(i) Let X and Y be sets from Rn. If X ⊂ Y then X is closed if
and only if Y is closed.
(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex
then either X or Y is closed and convex (one or the other).
(iii) Let X be an...

For each of the following statements, identify whether the
statement is true or false, and explain why. Please limit each
response to no more than 3 sentences.
i) A p-value is the probability that the null hypothesis is
false. ii) A chi-square test statistic can never be negative.
iii) If we reject the null hypothesis that a population
proportion is equal to a specific value, then that specific value
will not be contained in the associated confidence interval.
iv) If...

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...

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