# Determining whether two functions are linear independent via wronskian

• ysebastien
In summary, the conversation discusses determining the linear independence of two pairs of functions. The Wronskian is used to make this determination, and the calculations are shown for both pairs of functions. However, a mistake is made in the calculation for the second pair, leading to the incorrect conclusion that the functions are linearly independent. The correct calculation is provided and the conversation ends with a thank you for the clarification.
ysebastien

## Homework Statement

Determine which of the following pairs of functions are linearly independent.

(a) $f(t)=3t,g(t)=|t|$

(b) $f(x)=x^{2},g(x)=4|x|^{2}$

## Homework Equations

the Wronskian is defined as,

W=Det{{f(u),g(u)},{f'(u),g'(u)}}

if {f(u),g(u)} are linearly dependent, W=0

if W=/=0, {f(u),g(u)} are linearly independent

## The Attempt at a Solution

The assumed interval for the independent variables t,x are $x,t \in (-\infty,\infty)$

for (a),

I determined $W(t)=3t-3|t|$, which for x>0 is $W(t)=3t-3t=0$.
for x<0 we have $W(t)=-3t-3t=-6t$. Since for some value of $t \in (-\infty,\infty)$ I found a $W(t) \neq 0$ I can conclude that the functions f(t) and g(t) are linearly independent.

Now for (b),

Similarly to (a), I find a $W(x)=8x^{2}|x|-8|x|^{2}x$

for $x>0 : W(x)=8x^{3}-8x^{3}=0$
for $x<0 : W(x)=8(-x)^{2}|-x|-8|-x|^{2}(-x)=8x^{3}+8x^{3}=16x^{3}\neq 0$

Similarly I conclude that f(x) and g(x) are linearly independent since I found values of x which make the wronskian not equal to 0.

However, while my conclusion is correct for (a), (b) is supposedly linearly dependent.

Is my method correct? if so what mistake did I make in concluding that the functions of (b) were linearly independent?

Thanks

|x|^2 is equal to x^2.

Yes,

I use that fact, but I still find that for
$x<0 : W(x)=8(-x)^{2}|-x|-8(-x)^{3}=8x^{3}+8x^{3}=16x^{3}$

Am I just completely missing something here?

Your calculation of $W(x)=8x^{2}|x|-8|x|^{2}x$ is wrong. Just use that |x|^2=x^2 from the beginning, so g(x)=4x^2.

The specific place where your calculations are incorrect is$$\frac d {dx}|x|^2 = 2|x|$$is false.

Ah, I see. Thank you

## 1. What is the Wronskian?

The Wronskian is a mathematical tool used to determine whether two functions are linearly independent. It is denoted by the symbol W and is defined as the determinant of a matrix containing the derivatives of the two functions.

## 2. How is the Wronskian used to determine linear independence?

If the Wronskian of two functions is non-zero at a given point, then the two functions are linearly independent at that point. If the Wronskian is zero at all points, then the functions are linearly dependent.

## 3. Can the Wronskian be used to determine linear independence for more than two functions?

Yes, the Wronskian can be extended to determine linear independence for any number of functions. The general formula for the Wronskian of n functions is a(n) = det(W(f1, f2, ..., fn)), where a(n) is the Wronskian and f1, f2, ..., fn are the n functions.

## 4. What is the significance of the Wronskian in linear algebra?

The Wronskian is important in linear algebra because it can be used to determine the linear independence of a set of functions. This is useful in solving systems of linear differential equations, as well as in other areas of mathematics and physics.

## 5. Are there any limitations to using the Wronskian to determine linear independence?

Yes, there are limitations to using the Wronskian. It can only determine linear independence at a specific point, and it is not always accurate for functions with a complex or discontinuous behavior. Additionally, it cannot be used to determine linear independence for functions that are not differentiable.

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