Linear independent sets (1 Viewer)

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1. The problem statement, all variables and given/known data
determine whether the following are linear independent sets of vectors in the vector space R*R of all functions from R to R
a)fn:=1+t+......+t^n for n=1,.....4
b) sin,sin^2 ,sin^3
c)1,sin^2,cos^2

2. Relevant equations



3. The attempt at a solution
can i do like this for (b)(c)
sin^3=sin(sin^2)
sin^2+cos^2=1
so they are linearly dependent sets
 
That doesn't look like a linear combination.
why? I think sin^3=(sin) * (sin^2) is right
 
31,927
3,893
why? I think sin^3=(sin) * (sin^2) is right
It's right but completely irrelevant.

For b, use the definition of linear independence to show that the equation
c1*sin(t) + c2*sin2(t) + c3*sin3(t) = 0 has exactly one solution for the constants c1, c2, and c3. (If there is more than one solution, the functions are linearly dependent.)

For c, the functions 1, sin2(t) and cos2(t) are linearly dependent, as you said. The first function, 1, is a linear combination of the other two.

Another way to look at this is that the equation
c1*1 + c2*sin2(t) + c3*cos2(t) = 0 has a solution where not all of the constants are zero. One such solution is c1 = -1, c2 = 1, c3 = 1. There are lots of solutions.
 
It's right but completely irrelevant.

For b, use the definition of linear independence to show that the equation
c1*sin(t) + c2*sin2(t) + c3*sin3(t) = 0 has exactly one solution for the constants c1, c2, and c3. (If there is more than one solution, the functions are linearly dependent.)

For c, the functions 1, sin2(t) and cos2(t) are linearly dependent, as you said. The first function, 1, is a linear combination of the other two.

Another way to look at this is that the equation
c1*1 + c2*sin2(t) + c3*cos2(t) = 0 has a solution where not all of the constants are zero. One such solution is c1 = -1, c2 = 1, c3 = 1. There are lots of solutions.
i see thanks a lot
 

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