# find if the set is dependent or not?

• May 24th 2009, 03:57 PM
orendacl
find if the set is dependent or not?
The set is:
{1, cos(2x) 3*(sin(x))^2}

I figure that:
C1(1 +C2(cos(2x) + C3*(3*sin(x))^2 =0

That gives us:
C1(1) + C2(Cos(0) + C3(3(sin(0))^2 = 0

Thus C1 + C2 + 0 = 0
So:
C1 + C2 = 0

How do I find C3?

If my process is correct...Linear dependence exists if either C1 , C2 or C3 are non zero...
• May 24th 2009, 04:17 PM
arbolis
Wait, is the set $\{ 1, cos(2x) 3*(sin(x))^2\}$ as you wrote, or $\{ 1 , cos(2x) , 3*(sin(x))^2 \}$?
• May 24th 2009, 04:46 PM
orendacl
Hello, My apologies. Your comma is correct.
• May 24th 2009, 05:05 PM
ThePerfectHacker
Quote:

Originally Posted by orendacl
The set is:
{1, cos(2x) 3*(sin(x))^2}

Hint: $2\sin^2 x = 1 - \cos 2x$.
• May 24th 2009, 06:24 PM
orendacl
Hence one of the terms changes...

1, cos 2x, 1-cos3x thus our eq'n should look like:

c1(1) + c2(cos 2x) +c3(1-cos3x) = 0

That gives us
c1+c2=0 while c3=1?

Thus the solution is that
matrix looks like |1|
|0|
|0|

Thus the solution is dependent because one of the values is non-zero...
• May 24th 2009, 10:36 PM
Swlabr
Quote:

Originally Posted by orendacl
The set is:
...That gives us:
$C_1(1) + C_2(Cos(0) + C_3(3(sin(0))^2 = 0$...

You're trying to show that some linear combination of the three functions is zero, not that they pass through zero for a certain x. You're trying to show that there exist some $C_1, C_2, C_3 \in \mathbb{Z}$ such that $f(x) = C_1 +C_2*cos(2x) + C_3*(3*sin(x))^2$ and $f(x)=0$ for all $x$.

HINT: Look at ThePerfectHacker's post...
• May 27th 2009, 10:41 PM
orendacl
I still don't see it...
• May 28th 2009, 12:17 AM
Swlabr
Quote:

Originally Posted by orendacl
I still don't see it...

Don't think of the sine as functions, think of them as unknowns - call them $\alpha$, $\beta$ and $\gamma$: $\alpha = 1, \beta = cos(2x), \gamma = 3(sin(x))^2$. So, we want to find integers $C_1, C_2, C_3$ such that $C_1* \alpha + C_2* \beta + C_3 * \gamma = 0$, by the definition of linear dependence.

Substitute in the $\alpha, \beta, \gamma$ into TPH's equality and stick everything on one side. You should have something of the form $C_1* \alpha + C_2* \beta + C_3 * \gamma = 0$ and thus your set is not linearly independent.