Proving functions are linearly dependent

In summary, the conversation discusses a problem of proving linear independence of a set of functions in a linear space. The first three functions can be combined to add up to zero, but the fourth and fifth functions pose a challenge. It is mentioned that any two linearly dependent functions automatically make a larger set of vectors linearly dependent.
  • #1
brotherbobby
579
147
TL;DR 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##
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.
 
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  • #2
What about using the differentiability of these functions?
 
  • #3
brotherbobby said:
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
fresh_42 said:
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
fresh_42 said:
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
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).
 

1. What does it mean for functions to be linearly dependent?

Linear dependence refers to the relationship between two or more functions where one function can be expressed as a linear combination of the other functions. In other words, one function can be written as a constant multiple of another function.

2. How can I prove that functions are linearly dependent?

To prove that functions are linearly dependent, you can use the definition of linear dependence mentioned above. You can also use the method of substitution, where you substitute one function into another and see if it satisfies the equation. If it does, then the functions are linearly dependent.

3. What is the significance of proving functions are linearly dependent?

Proving that functions are linearly dependent is important in various fields of mathematics and science, including linear algebra, differential equations, and physics. It helps in understanding the relationships between different functions and can be used to simplify complex problems.

4. Can functions be linearly dependent in higher dimensions?

Yes, functions can be linearly dependent in higher dimensions as well. In fact, the concept of linear dependence is not limited to two or three dimensions, but it applies to any number of dimensions. In higher dimensions, the functions can be expressed as a linear combination of other functions in the same way as in two or three dimensions.

5. What is the difference between linearly dependent and linearly independent functions?

Linearly independent functions are those that cannot be expressed as a linear combination of other functions. In contrast, linearly dependent functions can be written as a linear combination of other functions. In other words, linearly independent functions are not related to each other, while linearly dependent functions have a relationship between them.

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