- #1
jeff1evesque
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Assignment question:
Let [tex]V = P (R)[/tex] and for j >= 1 define [tex]T_j(f(x)) = f^j (x)[/tex]
where [tex]f^j(x)[/tex] is the jth derivative of f(x). Prove that the
set {[tex]T_1, T_2,..., T_n [/tex]} is a linearly independent subset of [tex] L(V)[/tex]
for any positive integer n.
I have no idea how V= P(R) has anything to do with the rest of the problem- in particular with the transformation [tex]T_j(f(x)) = f^j (x)[/tex]. I guess I just don't understand how V ties in with the definition of the transformation T.
2. Relevant ideas:
I heard two different professors each saying a different method of proving this problem. One said it was possible to prove it by contradiction, and another tried to help me prove it directly.
So when I started the method of contradiction I had:
Let B = {[tex] a_1T_1, a_2T_2, ..., a_nT_n [/tex]}
Assume B is linearly dependent (goal: show a contradiction that its dependent?)
Choose [tex]T_i[/tex] from B such that 1<= i <= n.
Therefore, [tex]T_i = B - a_iT_i[/tex]
From here I couldn't continue. I have to show that the linear combination above (to the right of the equality) takes elements to the same ith derivative- and derive a contradiction here?
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The second method involve find the bases of V. A professor said it was {[tex]1, x/1!, x^2/2!, ..., x^n/n![/tex]}. But that doesn't make sense to me. I thought the basis would be {[tex]0, x, x^2,... x^n[/tex]}.
Let [tex]V = P (R)[/tex] and for j >= 1 define [tex]T_j(f(x)) = f^j (x)[/tex]
where [tex]f^j(x)[/tex] is the jth derivative of f(x). Prove that the
set {[tex]T_1, T_2,..., T_n [/tex]} is a linearly independent subset of [tex] L(V)[/tex]
for any positive integer n.
I have no idea how V= P(R) has anything to do with the rest of the problem- in particular with the transformation [tex]T_j(f(x)) = f^j (x)[/tex]. I guess I just don't understand how V ties in with the definition of the transformation T.
2. Relevant ideas:
I heard two different professors each saying a different method of proving this problem. One said it was possible to prove it by contradiction, and another tried to help me prove it directly.
So when I started the method of contradiction I had:
Let B = {[tex] a_1T_1, a_2T_2, ..., a_nT_n [/tex]}
Assume B is linearly dependent (goal: show a contradiction that its dependent?)
Choose [tex]T_i[/tex] from B such that 1<= i <= n.
Therefore, [tex]T_i = B - a_iT_i[/tex]
From here I couldn't continue. I have to show that the linear combination above (to the right of the equality) takes elements to the same ith derivative- and derive a contradiction here?
----------------
The second method involve find the bases of V. A professor said it was {[tex]1, x/1!, x^2/2!, ..., x^n/n![/tex]}. But that doesn't make sense to me. I thought the basis would be {[tex]0, x, x^2,... x^n[/tex]}.
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