Table of Contents

## Can an eigenvalue have no eigenvector?

(2) Every eigenvalue has at least one eigenvector. (3) Every eigenvector has ONLY one eigenvalue, that is, we cannot have two, or more, eigenvalues that correspond to the same eigenvector.

### What does it mean if 0 is an eigenvalue?

If 0 is an eigenvalue, then the nullspace is non-trivial and the matrix is not invertible. Therefore all the equivalent statements given by the invertible matrix theorem that apply to only invertible matrices are false.

#### Is 0 a distinct eigenvalue?

The distinct eigenvalues of A are 0,1,2. When eigenvalues are not distinct, it means that an eigenvalue appears more than once as a root of the characteristic polynomial.

**Can two different eigenvalues have the same eigenvector?**

The converse statement, that an eigenvector can have more than one eigenvalue, is not true, which you can see from the definition of an eigenvector. However, there’s nothing in the definition that stops us having multiple eigenvectors with the same eigenvalue.

**Are normalized eigenvectors unique?**

This is a result of the mathematical fact that eigenvectors are not unique: any multiple of an eigenvector is also an eigenvector! Different numerical algorithms can produce different eigenvectors, and this is compounded by the fact that you can standardize and order the eigenvectors in several ways.

## What is Normalised eigenvector?

3.4 Normalization of Eigenvectors it is straightforward to show that if |vŸ© is an eigenvector of A, then, any multiple N|vŸ© of |vŸ© is also an eigenvector since the (real or complex) number N can pull through to the left on both sides of the equation. Such an eigenvector is called normalized.

### Why do we normalize vectors?

Any vector, when normalized, only changes its magnitude, not its direction. Also, every vector pointing in the same direction, gets normalized to the same vector (since magnitude and direction uniquely define a vector). Hence, unit vectors are extremely useful for providing directions.

#### Are eigenvectors linearly independent?

Eigenvectors corresponding to distinct eigenvalues are linearly independent. As a consequence, if all the eigenvalues of a matrix are distinct, then their corresponding eigenvectors span the space of column vectors to which the columns of the matrix belong.

**How do you know if a vector 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.

**Does distinct mean linearly independent?**

Intuitively distinct means “all different”. Two vectors v1 and v2 are distinct if v2 – v1 is not zero. Consider v1 = (3 e1 + 4 e2) versus v2 = (6 e1 + 8 e2). They are distinct, yet they are not linearly independent since 2 v1 = v2.

## What are linearly independent vectors?

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.

### 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. Therefore v1,v2,v3 are linearly independent. Four vectors in R3 are always linearly 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. Any three linearly independent vectors in R3 must also span R3, so v1, v2, v3 must also span R3.

**Can 3 vectors span R4?**

Solution: A set of three vectors can not span R4. To see this, let A be the 4 Ã— 3 matrix whose columns are the three vectors. This matrix has at most three pivot columns. This means that the last row of the echelon form U of A contains only zeros.

**Can 4 vectors in R5 be linearly independent?**

FALSE. There are only four vectors, and four vectors can’t span R5.

## Can 4 vectors form a basis for R3?

Why? (Think of V = R3.) A basis of R3 cannot have more than 3 vectors, because any set of 4 or more vectors in R3 is linearly dependent.

### Can zero vector be a basis?

No. A basis is a linearly in-dependent set. And the set consisting of the zero vector is de-pendent, since there is a nontrivial solution to c†’0=†’0. If a space only contains the zero vector, the empty set is a basis for it.

#### Can a basis be infinite?

Every vector in a vector space can be written in a unique way as a finite linear combination of the elements in this basis. A basis for an infinite dimensional vector space is also called a Hamel basis.

**Can a vector space be infinite dimensional?**

The vector space of polynomials in x with rational coefficients. Not every vector space is given by the span of a finite number of vectors. Such a vector space is said to be of infinite dimension or infinite dimensional.

**Is Hilbert space infinite dimensional?**

Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane.