#Linear independence and generating of a family of functions

28 messages · Page 1 of 1 (latest)

haughty ridge
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This is my first time encountering a problem on a family of functions and I have no clue regarding the use of limits (as it is implied here I should use x near +inf) to prove linear independence (between f1 f2 and f3)

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little pebbleBOT
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little sigil
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start with definition of linear independence

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suppose a1 + bx + c*sinx = 0 for all x in R

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in particular for x=0, conclude that a = 0

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so now you're left with bx + csinx =0 for all x

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if you take limit as x tends to + inf, the expression is unbounded given that b is nonzero

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but the rhs is constantly 0, which is a contradiction

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so b=0 must hold

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and finally conclude that c=0 and you are done

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@haughty ridge

haughty ridge
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Thank you

little sigil
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of course

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you have to keep in mind the vectors are functions in this case

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equality of functions is defined pointwise

haughty ridge
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Oh yeah

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Well thanks I think that’s all i needed to solve the rest as well

little sigil
haughty ridge
little sigil
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yes

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with x=0 we get a=0

then put x = pi, then bpi + csinpi = bpi = 0, hence b = 0

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and finally c sin pi/2 = c = 0

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@haughty ridge

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you don't have to use the limit argument, but you can

haughty ridge
agile pollenBOT
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@haughty ridge

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