# Question: Are Sin 2x And Cos 2x Linearly Independent?

## How do you prove eigenvectors are linearly independent?

Let λ1, λ2, … , λk denote the distinct eigenvalues of an n × n matrix A with corresponding eigenvectors x1, x2, … , xk.

If all the eigenvalues have multiplicity 1, then k = n, otherwise k < n.

We use mathematical induction to prove that {x1, x2, … , xk} is a linearly independent set..

## Can 2 vectors in r3 be linearly independent?

If m > n then there are free variables, therefore the zero solution is not unique. Two vectors are linearly dependent if and only if they are parallel. … Four vectors in R3 are always linearly dependent. Thus v1,v2,v3,v4 are linearly dependent.

## 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 COSX and Sinx linearly independent?

a1cos(x)+a2sin(x)=θ(x)=0. If this linear combination has only the zero solution a1=a2=0, then the set {cos(x),sin(x)} is linearly independent.

## How do you determine if a set 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.

## Can 3 vectors in r4 be linearly independent?

No, it is not necessary that three vectors in are dependent. For example : , , are linearly independent. Also, it is not necessary that three vectors in are affinely independent.

## Can a non square matrix be linearly independent?

A square matrix is full rank if and only if its determinant is nonzero. For a non-square matrix with rows and columns, it will always be the case that either the rows or columns (whichever is larger in number) are linearly dependent.

## What are linearly independent solutions?

Is the set of functions {1, x, sin x, 3sin x, cos x} linearly independent on [−1, 1]? … Solution #1: The set of functions {1, x, sin x, 3sin x, cos x} is not linearly independent on [−1, 1] since 3sin x is a mulitple of sin x.

## 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.

## What does linearly independent mean in differential equations?

Definition: Linear Dependence and Independence. Let f(t) and g(t) be differentiable functions. Then they are called linearly dependent if there are nonzero constants c1 and c2 with c1f(t)+c2g(t)=0 for all t. Otherwise they are called linearly independent.

## Are trigonometric functions linear?

Trigonometric functions are also not linear. … The mistake is to assume that the function f(x) = cos(x) is linear, that is that f(x+y) = f(x) + f(y). A simple counterexample shows that this function f is not linear.

## How do you know if two functions 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 does it mean if two vectors are 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 cosine a linear system?

Also any function like cos(x) is non-linear. In math and physics, linear generally means “simple” and non-linear means “complicated”.

## Are Sinx and sin2x linearly independent?

Let f(x) = W(sin x,sin 2x). Then f(π/2) = 2(1)(−1) − 0 = −2, so sin x and sin2x are linearly independent. … Thus the functions are linearly dependent.

## Are the functions linearly independent?

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.