Question

Consider the following functions.

f1(x) = x, f2(x) = x-1, f3(x) = x+4

g(x) = c1f1(x) + c2f2(x) + c3f3(x)

Solve for c1, c2, and c3 so that g(x) = 0 on the interval (−∞, ∞).
If a nontrivial solution exists, state it. (If only the trivial
solution exists, enter the trivial solution {0, 0, 0}.)

{c1, c2, c3} =?

Determine whether f1, f2, f3 are linearly independent on the
interval (−∞, ∞).

linearly dependent or linearly independent?

Answer #1

Consider the following functions.
f1(x) = x, f2(x) =
x2, f3(x) = 6x −
4x2
g(x) = c1f1(x) +
c2f2(x) + c3f3(x)
Solve for
c1, c2,
and
c3
so that
g(x) = 0
on the interval
(−∞, ∞).
If a nontrivial solution exists, state it. (If only the trivial
solution exists, enter the trivial solution {0, 0, 0}.)
{c1, c2, c3} =

The functions
f1(x) = x
and
f2(x) = x6
are orthogonal on
[−4, 4].
Find constants
C1
and
C2
such that
f3(x) = x + C1x2 +
C2x3
is orthogonal to both
f1
and
f2
on the same interval.

Determine if the set of functions is linearly independent:
1. f1(x)=cos2x, f2(x)=1, f3(x)=cos^2 x
2. f1(x)=e^ x, f2(x)=e^-x, f3(x)=senhx

Determine whether the given functions are linearly dependent or
linearly independent.
f1(t) =
4t − 7,
f2(t) =
t2 + 1,
f3(t) =
6t2 − t,
f4(t) =
t2 + t + 1
linearly dependentlinearly independent
If they are linearly dependent, find a linear relation among them.
(Use f1 for f1(t),
f2 for f2(t),
f3 for f3(t), and
f4 for f4(t).
Enter your answer in terms of f1,
f2, f3, and
f4. If the system is independent, enter
INDEPENDENT.)

Consider the following predicate formulas.
F1: ∀x ( P(x) → Q(x) )
F2: ∀x P(x) → Q(x)
F3: ∃x ( P(x) → Q(x) )
F4: ∃x P(x) → Q(x)
For each of the following questions, answer Yes or No &
Justify briefly
. (a) Does F1 logically imply F2?
(b) Does F1 logically imply F3?
(c) Does F1 logically imply F4?
(d) Does F2 logically imply F1?

1. For the following, make sure you explain which basic
functions in F1-F3 you are using, and how exactly you are applying
the operations O1-O3.
(a) Show that every constant function g : N → N is recursive.
You may need to use induction.
(b) Show that the function f : N → N such that f(x) = x 3 is
primitive recursive.

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...

Write a Matlab script that plots the following functions over 0
≤ x ≤ 5π:
f1(x) = sin2 x − cos x,
f2(x) = −0.1 x 3 + 2 x 2 + 10,
f3(x) = e −x/π ,
f4(x) = sin(x) ln(x + 1).
The plots should be in four separate frames, but all four frames
should be in one figure window. To do this you can use the subplot
command to create 2 × 2 subfigures.

(a) Consider x^2 + 7x + 15 = f(x) and e^x = g(x) which are
vectors of F(R, R) with the usual addition and scalar
multiplication. Are these functions linearly independent?
(b) Let S be a finite set of linearly independent vectors {u1,
u2, · · · , un} over the field Z2. How many vectors are in
Span(S)?
(c) Is it possible to find three linearly dependent vectors in
R^3 such that any two of the three are not...

Find each of the following functions. f(x) = 4 − 4x, g(x) =
cos(x)
(a) f ∘ g and State the domain of the function. (Enter your
answer using interval notation.)
(b) g ∘ f and State the domain of the function. (Enter your
answer using interval notation.)
(c) f ∘ f and State the domain of the function. (Enter your
answer using interval notation.)
(d) g ∘ g and State the domain of the function. (Enter your
answer using...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 13 minutes ago

asked 16 minutes ago

asked 23 minutes ago

asked 33 minutes ago

asked 40 minutes ago

asked 43 minutes ago

asked 44 minutes ago

asked 50 minutes ago

asked 51 minutes ago

asked 1 hour ago

asked 1 hour ago