L(p(x)) = xp'(x)?
ok for this one I've got a_1 = 0 and a_2 = 0. So my answer is ker(L) = {0} But the real answer is span{1}.
What is Span{1}? I know what span is but not this. Please explain it to me in details. Thanks.
Answers & Comments
Verified answer
First of all,
ker L = {p(x): L(p(x)) = 0}, by definition of the kernel
........= {p(x): x p'(x) = 0}, by definition of L
........= {p(x): p'(x) = 0}
........= {a | a is any scalar}, since constants are the only functions with derivative equal to 0
........= span {1}.
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Note: span{f(x), g(x)} signifies all linear combinations of f(x) and g(x).
That is, span {f(x), g(x)} = {a f(x) + b g(x) | a, b are scalars}.
[This readily generalizes to more than 2 functions.]
In the case of one function, span {f(x)} = {a f(x) | a is any scalar}.
So, span {1} = {a * 1 = a | a is any scalar}, as indicated above
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I hope this helps!