Proving functions are linearly dependent

  • #1
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Summary:

We have set of functions ##\varphi (t)## continuous in the interval ##[a,b]##. The set is a linear (vector) space with the usual definitions of addition and multiplication by real numbers. We denote this space by ##C[a,b]##.

Statement of the problem : Prove that the following set of functions are linearly independent in the space ##C[a,b]## mentioned above : ##\varphi_1(t) = \sin^2 t, \varphi_2(t) = \cos^2 t, \varphi_3(t) = t, \varphi_4(t) = 3 \; \text{and} \; \varphi_1(t) = e^t##

Main Question or Discussion Point

We can make the first three functions add up to zero in the following way : ##\sin^2 t+\cos^2 t-\frac{1}{3}\times 3 = \varphi_1(t) + \varphi_2(t) - \frac{1}{3} \varphi_3(t) = 0##.

However, look at ##\varphi_4(t) = t## and ##\varphi_5(t) = e^t##. How does one combine the two to add up to zero? I can't see a way out.

Any help would be appreciated.
 

Answers and Replies

  • #2
PeroK
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What about using the differentiability of these functions?
 
  • #3
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Summary:: We have set of functions ##\varphi (t)## continuous in the interval ##[a,b]##. The set is a linear (vector) space with the usual definitions of addition and multiplication by real numbers. We denote this space by ##C[a,b]##.

##\text{Statement of the problem :}## Prove that the following set of functions are linearly independent in the space ##C[a,b]## mentioned above : ##\varphi_1(t) = \sin^2 t, \varphi_2(t) = \cos^2 t, \varphi_3(t) = t, \varphi_4(t) = 3 \; \text{and} \; \varphi_1(t) = e^t##

We can make the first three functions add up to zero in the following way : ##\sin^2 t+\cos^2 t-\frac{1}{3}\times 3 = \varphi_1(t) + \varphi_2(t) - \frac{1}{3} \varphi_3(t) = 0##.

However, look at ##\varphi_4(t) = t## and ##\varphi_5(t) = e^t##. How does one combine the two to add up to zero? I can't see a way out.

Any help would be appreciated.
You say you want to show linear independence, but you gave a non trivial linear combination?
If ##\vec{u},\vec{v}## are linearly dependent, then ##\{\,\vec{u},\vec{v},\vec{w}\,\}## are automatically linearly dependent, too.
 
  • #4
PeroK
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You say you want to show linear independence, but you gave a non trivial linear combination?
If ##\vec{u},\vec{v}## are linearly dependent, then ##\{\,\vec{u},\vec{v},\vec{w}\,\}## are automatically, too, linearly dependent.
I assumed the question is to investigate the linear indepence of these functions.
 
  • #5
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You say you want to show linear independence, but you gave a non trivial linear combination?
If ##\vec{u},\vec{v}## are linearly dependent, then ##\{\,\vec{u},\vec{v},\vec{w}\,\}## are automatically, too, linearly dependent.
Many thanks. Sorry I forgot this is a theorem. "Given any two linearly dependent functions, any larger set of vectors involving the two is also linearly dependent".

Many thanks and apologies.
 
  • #6
Infrared
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You don't need to cite any theorem: ##\sin^2(t)+\cos^2(t)+0\cdot t-\frac{1}{3}3+0\cdot e^t## is a nontrivial linear combination that is zero (and this example should make it clear why that theorem is true).
 

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