Table of Contents

## How do you prove two functions are linearly independent?

Let f and g be differentiable on [a,b]. If Wronskian W(f,g)(t0) is nonzero for some t0 in [a,b] then f and g are linearly independent on [a,b]. If f and g are linearly dependent then the Wronskian is zero for all t in [a,b]. Show that the functions f(t) = t and g(t) = e2t are linearly independent.

**How do you know if two functions are linearly dependent?**

Now, if we can find non-zero constants c and k for which (1) will also be true for all x then we call the two functions linearly dependent. On the other hand if the only two constants for which (1) is true are c = 0 and k = 0 then we call the functions linearly independent.

**How do you determine if vector functions are linearly independent?**

We have now found a test for determining whether a given set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero determinant. The set is of course dependent if the determinant is zero.

### How do you know if a differential equation is linearly independent?

This is a system of two equations with two unknowns. The determinant of the corresponding matrix is the Wronskian. Hence, if the Wronskian is nonzero at some t0, only the trivial solution exists. Hence they are linearly independent.

**How do you show a system is almost linear?**

A system is called almost linear (at a critical point (x0,y0)) if the critical point is isolated and the Jacobian at the point is invertible, or equivalently if the linearized system has an isolated critical point.

**How do you prove linear independence?**

If you make a set of vectors by adding one vector at a time, and if the span got bigger every time you added a vector, then your set is linearly independent.

#### Does E XY )= E x e y imply independence?

E(XY ) = E(X)E(Y ) is ONLY generally true if X and Y are INDEPENDENT. If X and Y are independent, then E(XY ) = E(X)E(Y ). However, the converse is not generally true: it is possible for E(XY ) = E(X)E(Y ) even though X and Y are dependent.

**How do you find the functions that are linearly dependent?**

Write down the following equation. If we can find constants c1 c 1, c2 c 2, …, cn c n with at least two non-zero so that (2) (2) is true for all x x then we call the functions linearly dependent.

**Can two linearly independent functions have a zero Wronskian?**

In fact, it is possible for two linearly independent functions to have a zero Wronskian! This fact is used to quickly identify linearly independent functions and functions that are liable to be linearly dependent. Example 2 Verify the fact using the functions from the previous example.

## What are the constants that make (2)(2) true for X?

If, on the other hand, the only constants that make (2) (2) true for x x are c1 = 0 c 1 = 0, c2 = 0 c 2 = 0, …, cn = 0 c n = 0 then we call the functions linearly independent. Note that unlike the two function case we can have some of the constants be zero and still have the functions be linearly dependent.

**How do you find the derivative of sin(x)?**

The derivative of sin(x) can be found from first principles. Doing this requires using the angle sum formula for sin, as well as trigonometric limits.