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.