## How can you prove 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.

## Are eigenvectors from the same eigenvalue linearly independent?

Eigenvectors corresponding to distinct eigenvalues are always linearly independent. It follows from this that we can always diagonalize an n × n matrix with n distinct eigenvalues since it will possess n linearly independent eigenvectors.

**How do you find the number of linearly independent eigenvectors?**

No of linearly independent Eigen vectors= no of distinct (different ) Eigen values. since in the above question the Eigen values would be (2,2) ,there was only one distinct Eigen value i.e, 2, therefore the no of independent Eigen vectors=1. if the Eigen values were (2,3) then the no of distinct Eigen value would be 2.

**What makes vector linearly independent?**

Given a set of vectors, you can determine if they are linearly independent by writing the vectors as the columns of the matrix A, and solving Ax = 0. If there are any non-zero solutions, then the vectors are linearly dependent. If the only solution is x = 0, then they are linearly independent.

### Is one vector linearly independent?

A set consisting of a single vector v is linearly dependent if and only if v = 0. Therefore, any set consisting of a single nonzero vector is linearly independent.

### Is the sum of 2 eigenvectors an eigenvector?

(3) The sum of two eigenvectors of a linear operator T is always an eigenvector of T. (4) Any linear operator on an n-dimensional vector space that has fewer than n distinct eigenvalues is not diagonalizable. 10 points Prove that similar matrices have the same characteristic polynomial and hence the same eigenvalues.

**How many linearly independent eigenvectors are there?**

two linearly independent eigenvectors

Since A is the identity matrix, Av=v for any vector v, i.e. any vector is an eigenvector of A. We can thus find two linearly independent eigenvectors (say <-2,1> and <3,-2>) one for each eigenvalue.

**Do all matrices have eigenvectors?**

Every real matrix has an eigenvalue, but it may be complex. It has eigenvectors if and only if it has eigenvalues, by definition. The Cayley-Hamilton theorem provides an easy characterization of whether a matrix has eigenvalues: the eigenvalues are exactly the roots of the characteristic polynomial.

## How to prove the linear independence of eigenvectors?

If there are no repeated eigenvalues (i.e., are distinct), then the eigenvectors are linearly independent . The proof is by contradiction. Suppose that are not linearly independent. Denote by the largest number of linearly independent eigenvectors. If necessary, re-number eigenvalues and eigenvectors, so that are linearly independent.

## How can you prove that vectors are not linearly independent?

The proof is by contradiction. Suppose that are not linearly independent. Denote by the largest number of linearly independent eigenvectors. If necessary, re-number eigenvalues and eigenvectors, so that are linearly independent. Note that because a single vector trivially forms by itself a set of linearly independent vectors.

**When does the spanning of an eigenvector fail?**

If there are repeated eigenvalues, but they are not defective (i.e., their algebraic multiplicity equals their geometric multiplicity), the same spanning result holds. However, if there is at least one defective repeated eigenvalue, then the spanning fails.

**How is an eigenvector related to a transformation?**

An eigenvector v of a transformation A is a vector that, when the transformation is applied to it, doesn’t change its direction, i.e., A v is colinear to v. The only thing that may change is its length. The factor of that change is its eigenvalue λ.