Prove that the list (x^3, sin(x), cos(x)) is linearly independent in V

In summary, the conversation discusses the proof that the list of functions (x^3, sin(x), cos(x)) is linearly independent in the vector space of real-valued functions. The approach of setting them as a linear combination and trying different values of x is discussed, with the conclusion that the functions are independent on the whole real line. The concept of derivatives is also brought up as a way to prove the independence.
  • #1
GluonZ
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0
Question: Prove that the list (x^3, sin(x), cos(x)) is linearly independent in V (V being the vector space of real-valued functions. In other words... common everyday math)

They're linearly independent, its pretty obvious. The issue is -- proving rigorously. This is not for an assignment, its for exam prep. (Friday)

Started by setting them as a linear combination:

c1*x^3 + c2*sin(x) + c3*cos(x) = 0 and trying to prove that ALL the coefficients need to be zero. But, nothing's rigorous enough.

Any ideas?
 
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  • #2
If that is the zero function, then for all x, the LHS is zero. Why don'y you try putting some values for x in? The most obvious choice shows that c_3 is zero, thence it is trivial to find a value of x impliying that c_1 is zero, and then c_2 has to be zero as well.
 
  • #3
Exactly. Although, instead of trying different values of x, I like staying with the easy x= 0:
Since these are functions
c1 x3+ c2 sin(x)+ c3 cos(x)= 0 means that it is 0 for all x.
Take x= 0 to get an easy equation. Of course it also follows that the derivative is 0 for all x. Differentiate and put x= 0 to get another easy equation. Differentiate a second time and set x= 0 to get a third easy equation.
 
  • #4
what is the domain of your functions? if only one point, then they are not independent.

on the whole real line, just look at their zeroes to see sin and cos are independent. also if f depends on sin and cos, then its derivatives depend on their derivatives. this gives a contradiction eventually.
 

1. How do you define linear independence?

Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of each other. In other words, no vector in the set can be expressed as a linear combination of the other vectors in the set.

2. How do you prove linear independence?

To prove linear independence, you need to show that the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is when all the coefficients (c1, c2, ..., cn) are equal to 0. This means that the only way to get a zero vector is by multiplying each vector by 0.

3. How does this apply to the list (x^3, sin(x), cos(x))?

In this case, we are working with a vector space V that contains functions. The list (x^3, sin(x), cos(x)) represents three different functions in V. To prove linear independence, we need to show that there is no non-zero solution to the equation c1x^3 + c2sin(x) + c3cos(x) = 0 for any values of c1, c2, and c3.

4. How do you prove that there is no non-zero solution?

One approach is to use the properties of the functions in the list to make a specific choice of values for c1, c2, and c3 that will result in the equation not being satisfied. For example, we can choose c1 = 1, c2 = 0, and c3 = -1. This would give us x^3 - cos(x) = 0, which is not true for all values of x. Therefore, the list is linearly independent.

5. Why is it important to prove linear independence?

Proving linear independence is important because it shows that a set of vectors or functions are essential and cannot be reduced to a linear combination of each other. This allows us to work with these vectors or functions more efficiently and accurately in mathematical and scientific applications.

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