Linear Combinations, Linear Independence

Second-order differential equations involve the second derivative of the unknown function (and, quite possibly, the first derivative as well) but no derivatives of higher order. For nearly every second-order equation encountered in practice, the general solution will contain two arbitrary constants, so a second-order IVP must include two initial conditions.

Given two functions y₁( x) and y₂( x), any expression of the form

where c₁ and c₂ are constants, is called a linear combination of y₁ and y₂. For example, if y₁ = e ^x and y₂ = x², then

are all particular linear combinations of y₁ and y₂. So the idea of a linear combination of two functions is this: Multiply the functions by whatever constants you wish; then add the products.

Example 1: Is y = 2 x a linear combination of the functions y₁ = x and y₂ = x²?

Any expression that can be written in the form

is a linear combination of x and x². Since y = 2 x fits this form by taking c₁ = 2 and c₂ =o, y = 2 x is indeed a linear combination of x and x².

Example 2: Consider the three functions y₁ = sin x, y₂ = cos x, and y₃ = sin( x + 1). Show that y₃ is a linear combination of y₁ and y₂.

The addition formula for the since function says

Note that this fits the form of a linear combination of sin x and cos x,

by taking c₁ = cos 1 and c₂ = sin 1.

Example 3: Can the function y = x³ be written as a linear combination of the functions y₁ = x and y₂ = x²?

If the answer were yes, then there would be constants c₁ and c₂ such that the equation

holds true for all values of x. Letting x = 1 in this equation gives

and letting x = −1 gives

Adding these last two equations gives 0 = 2 c₂, so c₂ = 0. And since c₂ = 0, c₁ must equal 1. Thus, the general linear combination (*) reduces to

which clearly does not hold for all values of x. Therefore, it is not possible to write y = x³ as a linear combination of y₁ = x and y₂ = x².

One more definition: Two functions y₁ and y₂ are said to be linearly independent if neither function is a constant multiple of the other. For example, the functions y₁ = x³ and y₂ = 5 x³ are not linearly independent (they're linearly dependent), since y₂ is clearly a constant multiple of y₁. Checking that two functions are dependent is easy; checking that they're independent takes a little more work.

Example 4: Are the functions y₁( x) = sin x and y₂( x) = cos x linearly independent?

If they weren't, then y₁ would be a constant multiple of y₂; that is, the equation

would hold for some constant c and for all x. But substituting x = π/2, for example, yields the absurd statement 1 = 0. Therefore, the above equation cannot be true: y₁ = sin x is not a constant multiple of y₂ = cos x; thus, these functions are indeed linearly independent.

Example 5: Are the functions y₁ = e ^x and y₂ = x linearly independent?

If they weren't, then y₁ would be a constant multiple of y₂; that is, the equation

would hold for some constant c and for all x. But this cannot happen, since substituting x = 0, for example, yields the absurd statement 1 = 0. Therefore, y₁ = e ^x is not a constant multiple of y₂ = x; these two functions are linearly independent.

Example 6: Are the functions y₁ = xe ^x and y₂ = e ^x linearly independent?

A hasty conclusion might be to say no because y₁ is a multiple of y₂. But y₁ is not a constant multiple of y₂, so these functions truly are independent. (You may find it instructive to prove they're independent by the same kind of argument used in the previous two examples.)

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