Question

Let y_1=sin(x) and y_2=cos(x). Which of the following functions are linearly independent from y_1, y_2?
A. sin^2(x)
B. 2cos(x)
C. sin(x)+cos(x)
D. 0
E. sin(2x)

Answer 1

B. 2 * y_2 (linearly dependent)
C. y_1 + y_2 (linearly dependent)
D. 0 * y1_1 (linearly dependent)
E. 2 * y_1 * y_2 (linearly dependent)

Answer 2

A. y_1 ^ 2 (NOT linearly dependent)

Answer 3

so a*sin(x)+b*cos(x)=0
where a and b are not both 0
if a =0, then b*cos(x)=0 and b can be any number since cos does equal 0 sometimes
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a=1 and b=1
so sin(x)+cos(x)=0 happens so this combination works (1,1)
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sin(2x)=2sin(x)cos(x)=sin(x)cos(x)+sin(x)cos(x)=0
so I'm thinking any of the above that can be written as a*sin(x)+b*cos(x) should work where a and b i guess can vary as it did in the sin(2x)
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so i think two of the sets are linear dependent
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I could be wrong though

Answer 4

a*sin(x)+b*cos(x)=0
where a and b are not all 0
means {sin(x),cos(x)} is linearly independent

Answer 5

but I'm thinking a and b don't exactly have to be constant numbers as shown in my sin(2x)

Answer 6

just thinking

Answer 7

oops got my definitions turned around

Answer 8

a*sin(x)+b*cos(x)=0
where a and b are not all 0
means {sin(x),cos(x)} is linearly dependent
so i think two of the sets are linear independent *

Answer 9

expressions not sets*

Answer 10

I am not sure of my answers.

Answer 11

I think all of your answers are good
I'm just iffy on the 0 there

Answer 12

I was curious about the 0 also..

Answer 13

@vmarroquin1002 Are you allowed to pick more than one answer?

Answer 14

Two vectors u and v are linearly independent if the only numbers x and y satisfying xu+yv=0 are x=y=0. If we let

Answer 15

Yes I am. Sorry I forgot to clarify that.

Answer 16

this was copied from http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/lindep/lindep.html

Answer 17

so according to that definition all of the things can be zero for the definition linearly independent to apply

Answer 18

so @vmarroquin1002 are you cool? like do you feel you need any further explanation on anything?

Answer 19

actually i'm still thinking about these answers

Answer 20

It was A and E

Answer 21

What is the definition for linear independence given in your book/notes?

Answer 22

a*sin(x)+b*cos(x)
I guess that means a and b have to be constants and can all be zero?

Answer 23

Yeah, that is what it looks like.

Answer 24

using my logic from earlier
i could have said sin^2(x)=sin(x)*sin(x)+0*cos(x) lol

Answer 25

the only one that wouldn't work was the 0
since a and b couldn't both be 0

Answer 26

from my logic earlier

Answer 27

I wasn't even thinking vectors. I was trying to express each choice as a combination of y_1 and y_2 without raising either one to a power (other than 1).