1)I would like to show that T : F ? R be defined by T(f) = f^2(1) is not a linea
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1)I would like to show that T : F ? R be defined by T(f) = f^2(1) is not a linear transformation, I know I have to show that it doesn't preserve the addition or preserve scalar multiplication but i don't know how to write it. 2)Let C0([0, 1]) be the set of continuous, real-valued function on the closed interval [0, 1]. Let T : C0([0, 1]) ? R be the linear transformation defined by T(f) = integral from 0 to 1 of f(x) dx Find a linearly independent set of three functions in ker(T). 3)Let C^0 ([0, 2]) be the set of continuous, real-valued functions on the closed interval [0, 2pi]. Define an inner product on this space by = integral from 0 to 2pi f(x)g(x) dx. Prove that sin(x) and cos(x) are orthogonal functions with this inner product. Then find one more continuous function that is orthogonal to both of these (but not the constant function 0). I know inner product is same as dot product and to show it orthogonal we need to have the A*B=0 .. but i don't know how to write it... any help will be apreciatedExplanation / Answer
1) Let f be the identity function. Then T(2f) = 4 =/= 2 = 2T(f), so T is not linear.
2) Just find three reasonably different functions in ker(T), and they'll almost certainly be linearly independent. (I'm cheating somewhat with the word "reasonably", but common sense should suffice here.) For example, it's easy to check that polynomials are linearly independent, so try the functions:
f(x) = 2x-1
g(x) = 3x^2 - 1
h(x) = 4x^3 - 1
To check linear independence explicitly, suppose af + bg + ch=0. The x-coefficient is 2a, so a=0. Similarly, b=c=0.
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