Question

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

Answer #1

Note: we check af1(x)+bf2(x)+cf3(x)=0 if all a, b, c, are zero then linearly independent otherwise linearly dependent.

Details explained in the image.

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 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?

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.

Prove the following identities. (a) F1 +F3 +F5 +...+F2n−1 = F2n.
(b) F0 −F1 +F2 −F3 +...−F2n−1 +F2n = F2n−1 −1. (c) F02 +F12 +F2
+...+Fn2 = Fn ·Fn+1. (d) Fn−1Fn+1 − Fn2 = (−1)n.
Discrete math about Fibonacci numbers

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 textile mill model -
Produces 3 types of fabric (F1, F2, F3) on 2 types of machines
(Regular, Special). Or Buy.
Time to delivery is 13 weeks.
15 regular machines (F2, F3) + 3 special machines (F1, F2, F3)
+ Buy (F1, F2, F3)
Demand for F1, F2, F3 is 45000, 76500, 10000 yards.
Special loom capacity in yards/hour – 4.7, 5.2, 4.4
Regular loom capacity in yards/hour – 0, 5.2, 4.4
Manufacture cost in $/yard – 0.65, 0.61,...

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.

1.Show that cos 2t, sin 2t, and e^5t are linearly independent
and form a fundamental set of solutions for the equation: y ′′′ −
5y ′′ + 4y ′ − 20y = 0
2.Find the general solution to the equation: y ′′′ − y ′′ − 4y ′
+ 4y = 0

Show whether the following vectors/functions form linearly
independent sets:
(a) 2 – 3x, x + 2x^2 , – x^2 + x^3
(b) cos x, e^(–ix), 3 sin x
(c) (i, 1, 2), (3, i, –2), (–7+i, 1–i, 6+2i)

Three factories F1, F2 and F3 respectively produce 25%,
35% and 40% of the total number of electrical parts intended for
the assembly of a machine. These factories respectively produce 1%,
2% and 3% of defective parts.
We notice : The event A : "the part is produced by the F1
factory"
The event B : "the part is produced by the F2
factory"
The event C : "the part is produced by the F3 factory"
The event D : "the...

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