Are These Vectors Linearly Independent in R*R?

[ak123456]

Homework Statement

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

Homework Equations

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

 

[wywong]

sin^3=sin(sin^2)

That doesn't look like a linear combination.

 

[ak123456]

That doesn't look like a linear combination.

why? I think sin^3=(sin) * (sin^2) is right

 

[Mark44]

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.

 

[ak123456]

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