Linear combination of functions -- meaning?

[Vol]

h(x) = cf(x) + kg(x) is the linear combination of functions. What makes it linear?

 

[DrClaude]

https://en.wikipedia.org/wiki/Linear_combination:

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

 

[fresh_42]

h(x) = cf(x) + kg(x) is the linear combination of functions. What makes it linear?

The absence of any operation other than addition and scalar multiplication.

 

[WWGD]

h(x) = cf(x) + kg(x) is the linear combination of functions. What makes it linear?

Do you mean to ask whether the expression cf(x)+kg(x) is linear or whether h(x) is linear ( It is not necessarily linear)?

 

[Mark44]

Do you mean to ask whether the expression cf(x)+kg(x) is linear or whether h(x) is linear ( It is not necessarily linear)?

Here's what the OP wrote:

h(x) = cf(x) + kg(x) is the linear combination of functions.

I believe he was asking about the meaning of the expression "linear combination," and not whether either of the constituent functions was linear.

 

[RPinPA]

One answer is that it fits the definition of "linear combination", which @DrClaude gave you at #2. But are you asking why the terminology is used, why it is called a linear combination?

I don't know the history of the term. I do know the expression is linear in the parameters $c$ and $k$, neither appears with an exponent greater than 1. Because of that if you are doing curve fitting to this form, trying to find the optimal values of $c$ and $k$ for a given fixed $f(x)$ and $g(x)$, then you use linear least squares. Exactly the same procedure as fitting a straight line.

 

[DrClaude]

I don't know the history of the term. I do know the expression is linear in the parameters $c$ and $k$, neither appears with an exponent greater than 1. Because of that if you are doing curve fitting to this form, trying to find the optimal values of $c$ and $k$ for a given fixed $f(x)$ and $g(x)$, then you use linear least squares. Exactly the same procedure as fitting a straight line.

What you wrote is incorrect. It is the fact that $f(x)$ and $g(x)$ appear with to the power of 1 that makes it a linear combination. What you wrote about $c$ and $k$ doesn't make sense:
$$
h(x) = cf(x) + w^2 g(x)
$$
is as much a linear combination as
$$
h(x) = cf(x) + k g(x)
$$
(Hint: set $k \equiv w^2$.)

 

[RPinPA]

What you wrote about $c$ and $k$ doesn't make sense:

Nevertheless, that's the definition of linear least squares. For instance, linear least squares includes the problem of fitting of polynomials $f(x) = \sum_{k=0}^n a_k x^k$. The model is linear in the parameters The least squares criterion leads to a set of linear equations in the $a_k$.

I do know the expression is linear in the parameters $c$ and $k$

As it is. That's a correct statement.

$h(x)=cf(x)+w^2g(x)$

That doesn't negate my comments about linear least squares. If you were trying to find the optimum parameters, then using this model would not allow you to use linear least squares. The model is not linear in these chosen parameters. The equations for an optimal value of $c$ and $w$ would not be linear in those parameters and could not be solved with linear algebra methods. But as you note, you can change it to a linear model.

An example of a nonlinear model would be $f(x) = a e^{bx} + c$. That is nonlinear in $x$, but what makes it nonlinear least squares is the fact that it's nonlinear in $b$. The model $f(x) = a e^{3x} + c$ would allow use of linear least squares methods.

My comments are correct. Including the comment that I didn't know the history of the term including why it was applied here. That's a correct statement. I accept that my comments, while correct as to linear least squares, are not relevant to OP's question. But they are correct :-)

 

[fresh_42]

do know the expression is linear in the parameters ccc and kkk, neither appears with an exponent greater than 1.

This is the answer to a question which wasn't posed. Furthermore it is definitely wrong. As you can see, the LHS of $h(x) = cf(x) + kg(x)$ depends on $x$ and does not depend on neither $c$ nor $k$ of the RHS. This makes $c,k$ scalars. To implicitly assume such a dependency, despite it is explicitly ruled out, is a misinformation here and yes, wrong.

The question is not: what is a linear dependency, the question is, what is a linear combination. For the latter it is completely irrelevant what you wrote, i.e. which powers the scalars are written in! $h(x) = cf(x) + k^2g(x)$ is as a linear combination as $h(x) = cf(x) + kg(x)$ is!

 

[Mark44]

Thread closed, as the question has been asked and answered. @Vol, if you are still not certain, please send me a PM and I will reopen this thread.