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I've noticed that the term gets abused alot. For instance, suppose I have

$c_1 x_1 + c_2 x_2 = f(x)$...(1)

Eqtn (1) is such what we say "a linear combination of $x_1$ and $x_2$"

In ODE, sometimes when we want to solve a homogeneous 2nd order ODE like $y'' + y' + y = 0$, we find the characteristic eqtn and solve for the roots and put it into whatever form necessary. But in all casses, the solution takes form of $c_1y_1 + c_2y_2 = y(t)$.

The thing is that $y_1$ and $y_2$ itself doesn't even have linear terms, so does it make sense to say $c_1y_1^2 +c_2y_2^2 = f(t)$ is a "quadratic" combination of y_1 and y_2?

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I'm not sure what your last paragraph actually says. Could you clarify? –  Steven Taschuk Mar 14 '12 at 19:07
    
No because $+$ is a linear operator, thus a linear combination. –  Josué Molina Mar 14 '12 at 19:09
    
$\times {\rm constant}$ is a linear operations ($(x+\Delta x)\times c = x\times c + \Delta x \times c$). Same for $+$ is a linear operator. $c_1y_1^2 +c_2y_2^2$ is a linear combination of $y_1^2$ and $y_2^2$. A linear combination of $x$ and $y$ is $c_1 x + c_2 y$ where $c_1, c_2$ are constants w.r.t. $x$ and $y$. –  user2468 Mar 14 '12 at 19:13
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4 Answers

up vote 9 down vote accepted

You ask "What exactly do we mean when say “linear” combination?"

A linear combination is an expression of the form "(scalar times object ) + (scalar times object) + ... + (scalar times object)".

Of course for such an expression to make sense you need your objects to be amenable to be multiplied by scalars and added: that's why the usual context is a vector space.

And I strongly disagree with "the term gets abused a lot": I have never seen it abused, and I cannot even imagine how it could be abused. It is among the less ambiguous terms in math.

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Saying $c_1y_1^2+c_2y_2^2$ is quadratic in $y_1$ and $y_2$ would make sense to most people, but your talking about terms that the $y$s "have" leads me to believe you are approaching the meaning of the operative phrase too reductionistically. A skyscraper will involve putting flat materials together at right angles, but the cross-beams will not have such a linear structure at, say, the molecular level.

A linear combination of objects is a form (expression) that is linear "in" those objects - meaning that if we take the objects to be independent variables we could control at will, the form would vary linearly according to each of their changes. Thus $c_1x_1+\cdots c_nx_n$ is the general (finite) form.

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It's a linear combination in the vector space of continuous (or differentiable or whatever) functions. $y_1$ and $y_2$ are vectors (that is, elements of the vector space in question) and $c_1$ and $c_2$ are scalars (elements of the field for the vector space, in this case $\mathbb{R}$). In linear algebra it does not matter what kind of elements vector spaces consists of (so these might be tuples in the case of $\mathbb{R}^n$, or linear operators, or just continuous functions, or something entirely different), but that vector spaces satisfy the axioms which are algebraic laws.

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I am sure about the linear combination like ODE

In which equation on power of variable involve is and multiplied by any scalar quantity. Such combination is known as linear combination of algebra

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Your answer is hard to understand, perhaps because you are not a native English speaker? Is your first sentence meant to read "I am un-sure about the linear combination like ODE"? –  user1729 Nov 5 '13 at 11:41
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