are the functions 1+x,1-x,1+x+xsquare linearly independent or linearly
dependent? why?

From mathworld.com :

"The n functions f1(x), f2(x), ..., fn(x) are linearly dependent if,
for some c1, c2, ..., cn not all zero, sum(ci*fi) = 0"

In your question you have :

f1(x) = 1+x
f2(x) = 1-x
f3(x) = 1+x+x²

So if f1, f2 and f3 are linearly dependent, we can find constants a, b
and c such that :

a*f1(x) + b*f2(x) + c*f3(x) = 0 for every imaginable value of x.

AND either a <> 0, b <> 0 or c <> 0.

We can expand out the equation to get :

a*f1(x) + b*f2(x) + c*f3(x) = a + ax + b - bx + c + cx +cx² = 0

We can collect together terms :

(a + b + c) + (a - b + c)x + cx² = 0

For this to be true for all x, we need to have each coefficient for
each power of x = 0.

So straight away, we see that c = 0. (Otherwise we have an
"uncontrollable" term in x²)

Then a + b = 0 and a - b = 0. This can only be true if b = 0.

This implies that a = 0. But we need to have at least one of a, b or c
be non-zero. Hence f1, f2 and f3 cannot be linearly dependent.

So they must be linearly independent.

Hope this helps

calebu2-ga

Useful resources :
http://mathworld.wolfram.com/LinearlyDependentFunctions.html

Search strategy :
Linearly dependent functions

To show that f1,f2, f3 are linearly indep, i.e. to prove a=b=c=0
 From
 (a + b + c) + (a - b + c)x + cx² = 0 ...(*)
 Since x is arbitrary,
 Set x = 0, we have a + b + c = 0..(1)
 Take derivative of (*): a - b + c + 2cx = 0..(**)
 Set x = 0, we have a - b + c = 0..(2)
 Set x = 1, we have a - b + 3c = 0..(3)
 (3)-(2): c = 0 so we have a + b = 0 & a - b = 0 
 Hence, a = b =c = 0

 I don't suggest that assigning c = 0 to start solving the equation.
Even
 theoretically it is OK. (about an  "uncontrollable" term in x²)
 
A general consideration:
We also can give a simple explanation why they are linearly indep:
 Because the determinant of the 3x3 coefficient matrix is non-zero.
 |1  1 0|
 |1 -1 0| (det = -2)(since {1, x,x^2} is a basis of the vs P2 (poly of
deg <=2)
 |1  1 1|
 
 In fact, if f1= a1 + b1x+ c1 x^2
             f2= a2 + b2x+ c2 x^2
             f3= a3 + b3x+ c3 x^2
 if the det of the coefficient matrix is non-zero then f1,f2, f3 must
be linearly indep. In other words, we don't have to test the indep one
by one.


 Kenny