Linear dependence of functions

In summary: If the only solution is ##a_1 = 0## and ##a_2 = 0##, you will have proved that the functions are linearly independent. If you get some other solution, you will have proved that the functions are linearly dependent.As you get further into linear algebra, you'll start to see the same concepts applied to more abstract structures, like linear transformations of vector spaces, and you'll see how linear independence/dependence is really a statement about the structure of a vector space, and not about a particular equation.There is basically just one way to show linear independence (not independency), and that is to show that the equation ##c_1f_1(x) + c_2f_2
  • #1
Dank2
213
4

Homework Statement


check for linear dependecy[/B]
f(x) = cosx and g(x) = xcosx
2 functions from R to R

Homework Equations

The Attempt at a Solution


Why this is wrong:
if i take the scalar a1 = 3, a2 = 1
i can do that since 3 is real, and a1 is in R.
so 3f(3) + -1g(3) = 0
there for we have none trivial comb for the zero vector.
 
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  • #2
Dank2 said:

Homework Statement


check for linear dependecy[/B]
f(x) = cosx and g(x) = xcosx
2 functions from R to R

Homework Equations

The Attempt at a Solution


Why this is wrong:
if i take the scalar a1 = 3, a2 = 1
i can do that since 3 is real, and a1 is in R.
so 3f(3) + -1g(3) = 0
there for we have none trivial comb for the zero vector.
No.
You need to find scalars ##c_1## and ##c_2## for which ##c_1\cos(x) + c_2x\cos(x) = 0## for all real numbers x.

Here you have one equation with two unknowns, ##c_1## and ##c_2##. The usual trick to get another equation is to take the derivative of the first equation
 
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  • #3
Mark44 said:
No.
You need to find scalars ##c_1## and ##c_2## for which ##c_1\cos(x) + c_2x\cos(x) = 0## for all real numbers x.

Here you have one equation with two unknowns, ##c_1## and ##c_2##. The usual trick to get another equation is to take the derivative of the first equation
i can use here x =4
then a1 = 4 = 3 ===> a1 = 0
if a1 =0, then a2 =0 for the equation to hold.

is that ok
 
  • #4
Dank2 said:
i can use here x =4
No. Did you understand what I wrote in my previous post?
You need to find scalars ##c_1## and ##c_2## for which ##c_1\cos(x) + c_2x\cos(x) = 0## for all real numbers x.
Dank2 said:
then a1 = 4 = 3 ===> a1 = 0
if a1 =0, then a2 =0 for the equation to hold.

is that ok
No it's not. You wrote "a1 = 4 = 3" - what is this?
Clearly a1 = 0 and a2 = 0 is a solution whether the functions are linearly dependent or linearly independent. The deciding factor is whether there is a non-trivial solutions for these constants, independent of the value of x. In other words, don't choose a value for x.

I also gave a suggestion in my previous post.
 
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  • #5
If the trivial equation needs to be hold for all X, then for both X =0 it should hold and X = Pi.

For X= 0 we get:

a1f(X) + a2f(X) = 0
==> 0 + a2 = 0 ==> a2 = 0
Now for X=Pi we get:
-Pi*a1 + a2 = 0 , but a2 =0. So the fore a1 = 0 .

We need to find single pair that holds the trivial equation for all X, then I took two particular x's and got that both of a1 and a2 has to be zero. Is that right?
 
Last edited:
  • #6
Dank2 said:
If the trivial equation needs to be hold for all X, then for both X =0 it should hold and X = Pi.

For X= 0 we get:

a1f(X) + a2f(X) = 0
This will work, but you have two functions, not one.
The equation is ##a_1\cos(x) + a_2x\cos(x) = 0##
Here f(x) = cos(x) and g(x) = xcos(x).
If x = 0, what does the equation above simplify to?
Dank2 said:
==> 0 + a2 = 0 ==> a2 = 0
Now for X=Pi we get:
-Pi*a1 + a2 = 0 , but a2 =0. So the fore a1 = 0 .
Substitute x = ##\pi## into the equation ##a_1\cos(x) + a_2x\cos(x) = 0##
Dank2 said:
We need to find single pair that holds the trivial equation for all X, then I took two particular x's and got that both of a1 and a2 has to be zero. Is that right?
It's not the equation that is trivial -- the trivial solution is ##a_1 = 0, a_2 = 0##. If the equation has only the trivial solution, the functions are linearly independent. If the equation has solutions in addition to the trivial solution, the functions are linearly dependent.
 
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  • #7
Dank2 said:
a1f(X) + a2f(X) = 0
Mark44 said:
This will work, but you have two functions, not one.
yeah, i meant a2g(x).
so it will simplify as above. besides that solution, what other ways i can show their linear independency?
i mean without using values for x.
 
  • #8
Dank2 said:
yeah, i meant a2g(x).
so it will simplify as above. besides that solution, what other ways i can show their linear independency?
i mean without using values for x.
There is basically just one way to show linear independence (not independency), and that is to show that the equation ##c_1f_1(x) + c_2f_2(x) + \dots c_nf_n(x) = 0## has exactly one solution for the constants ##c_1, c_2, \dots, c_n##; namely all of them being zero. This definition is very subtle for many beginning students in this area, because ##c_1 = c_2 = \dots = c_n = 0##; is always a solution, whether the functions are linearly independent or linearly dependent. The deciding factor is whether there are solutions other than the trivial solution (all constants equat to zero).

Almost exactly the same equation and idea applies to linearly independent/dependent vectors.

One shortcut you can take: if you have two functions or two vectors, the two are linearly independent if neither one is a constant multiple of the other. Once you have three or more functions/vectors, you can't tell as easily.

Going back to your original work, with ##a_1\cos(x) + a_2x\cos(x) = 0##, since this equation has to be true for all values of x, it has to be true for two values you choose, so you can substitute two different values of x into it to get two different equations. From these equations you can solve for the constants ##a_1## and ##a_2##.
 
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What is linear dependence of functions?

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

How can you determine if functions are linearly dependent?

To determine if functions are linearly dependent, you can use the method of substitution. This involves substituting one function into another and seeing if the resulting equation is true for all values of the independent variable. If the equation is always true, then the functions are linearly dependent.

What is the importance of understanding linear dependence of functions?

Understanding linear dependence of functions is important in many areas of mathematics and science. It helps in solving systems of equations, finding solutions to differential equations, and understanding the behavior of complex systems.

Can linearly dependent functions be graphed?

Yes, linearly dependent functions can be graphed. However, the graph of a linearly dependent function will not show any new information as it will simply be a straight line that overlaps with another function.

What are some real-life applications of linear dependence of functions?

Linear dependence of functions is used in various fields such as physics, engineering, economics, and statistics. Some examples include modeling the relationship between force and displacement in a spring, analyzing the relationship between supply and demand in economics, and predicting the growth of a population over time.

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