# How Do You Show That A Solution Is Linearly Independent?

## How do you prove a function is linearly independent?

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.

We compute the Wronskian..

## How do you determine if something is 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.

## What does it mean to be linearly independent?

In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent.

## Is 0 linearly independent?

The following results from Section 1.7 are still true for more general vectors spaces. A set containing the zero vector is linearly dependent. A set of two vectors is linearly dependent if and only if one is a multiple of the other. A set containing the zero vector is linearly independent.

## Are linearly independent if and only if?

A set of two vectors is linearly independent if and only if neither of the vectors is a multiple of the other. A set of vectors S = {v1,v2,…,vp} in Rn containing the zero vector is linearly dependent. Theorem If a set contains more vectors than there are entries in each vector, then the set is linearly dependent.

## Are sin 2x and cos 2x linearly independent?

Since a and b are constants, but cos2(x) varies with x with 0≤cos2(x)≤1, the equation in (1) can only always be true only if b−a=0, so then a=0 also, resulting in b=0. Thus, this shows sin2(x) and cos2(x) are linearly independent.

## What is independent function?

Noun. independent function (plural independent functions) (mathematics) Any of a set of functions the value of which can not be derived from that of all the others.

## How do you know if two solutions are linearly independent?

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.

## What if the wronskian is zero?

If f and g are two differentiable functions whose Wronskian is nonzero at any point, then they are linearly independent. … If f and g are both solutions to the equation y + ay + by = 0 for some a and b, and if the Wronskian is zero at any point in the domain, then it is zero everywhere and f and g are dependent.

## Can 4 vectors in r3 be linearly independent?

The dimension of R3 is 3, so any set of 4 or more vectors must be linearly dependent.

## What are linearly independent functions?

One more definition: Two functions y 1 and y 2 are said to be linearly independent if neither function is a constant multiple of the other. For example, the functions y 1 = x 3 and y 2 = 5 x 3 are not linearly independent (they’re linearly dependent), since y 2 is clearly a constant multiple of y 1.