Question

Show that every vector in the \(N\) -dimensional vector space \(V^{*}\) has \(N-1\) linearly independent annihilators. Stated differently, show that a linear functional maps \(N-1\) linearly independent vectors to zero.

Short answer

In general, for an \(N\)-dimensional vector space \(V^{*}\), for each vector in it, there exist \(N-1\) linearly independent vectors that are all annihilated by the functional of the vector (mapped to zero). This is due to the fact that no non-trivial linear combination of these vectors equals the zero vector.

Step-by-step solution

Understand the defintions

The fundamental step involves understanding the terminologies involved in the exercise. A vector space is a set of vectors that follow specific rules. Annihilators are a set of all vectors that get mapped to zero under a given linear transformation. In this context, a linear functional is a function that maps a vector to a scalar, but follows the rules of linearity. Two vectors are linearly independent if none of them can be represented as a linear combination of the other.

Determine a generic vector function and its functional

Consider an \(N\)-dimensional vector space \(V\) and a generic vector \(v\) within that space having \(N\) components, say \(v_1, v_2, ..., v_N\). Similarly, denote an arbitrary linear functional in the dual space \(V^{*}\) as \(f\). Thus, \(f(v) = f_1v_1 + f_2v_2 + ... + f_Nv_N\). Here, \(f_i\) are the components of the functional.

Annihilator mapping to zero

The definition of the annihilator states that it gets mapped to zero. So if we consider \(N-1\) vectors, then the sum of their mappings under \(f\), \(f(v_1v) + f(v_2v) + ... + f(v_{N-1}v) = 0\). This is the fundamental property of annihilators that we need to use in the proof.

Proving the vectors are linearly independent

We have to prove that these \(N-1\) vectors are linearly independent. This can be done by showing that no non-trivial linear combination of these vectors can be equal to the zero vector. Here, the property of annihilators used in Step 3 will help us.

Key concepts

Vector Spaces

A vector space is a foundational concept in linear algebra that consists of a set of elements called vectors. In a vector space, these vectors follow specific rules, such as being able to be added together or scaled by numbers, called scalars. Vectors in a vector space work just like arrows. You can add them by placing the tail of one at the head of the other and rescale them by changing their length. The rules of vector space must follow certain axioms:

• Closure under addition and scalar multiplication: Adding any two vectors in the space results in another vector within the space. Similarly, multiplying a vector by a scalar results in another vector in the space.
• Associativity and commutativity: Vector addition is both associative and commutative.
• Existence of additive identity and inverses: There is a zero vector in the space that, when added to any vector, does not change it. Also, for every vector, there is another vector that when added together gives the zero vector.
• Distributive properties: A scalar scaling factor can distribute over vector addition, and the reverse is true for a vector distributing over scalar addition.

These properties make vector spaces a useful framework for studying mathematical and physical systems.

Linear Functional

A linear functional is a specific type of function that maps a vector from a vector space to a scalar field, like the real numbers \(\mathbb{R}\) or complex numbers \(\mathbb{C}\). However, unlike other functions, linear functionals maintain linearity, which means they adhere to these properties:

• Additivity: For vectors \(u\) and \(v\), the functional \(f\) satisfies \(f(u + v) = f(u) + f(v)\).
• Homogeneity of degree 1: For any scalar \(c\) and vector \(v\), \(f(cv) = cf(v)\).

This means linear functionals are simple yet powerful tools in analyzing the properties of vector spaces, especially because they can easily transform vectors into simpler numerical forms while preserving information about their structure. These mappings are essential in applications like physics and engineering, where analyzing projections and forces often involves using linear functionals.

Linearly Independent Vectors

Vectors are said to be linearly independent when no vector in the set can be written as a combination of others. This is a crucial concept in linear algebra because it helps define the dimension of a vector space, which is the number of vectors in the largest independent set.

To determine linear independence:

• You start by forming a linear combination of vectors, say \(c_1v_1 + c_2v_2 + ... + c_nv_n = 0\).
• For the combination to only produce the zero vector if all constants \(c_1, c_2, ..., c_n\) are zero, the vectors must be independent.

If even one vector can be expressed as a linear combination of others, the set is dependent, meaning more vectors than necessary are describing the space. This depends on how vectors in spaces interact through addition or scaling.

Annihilators

Annihilators in linear algebra refer to a collection of vectors that, when operated on by a linear functional, result in zero. Imagine a vector space \(V\) and its dual space \(V^*\), where functionals reside. An annihilator of a vector space is essentially a set of all such functionals that kill (or map to zero) every vector in a subset of \(V\).

The concept finds its significance in the study of linear transformations and system solutions. Specifically, if you have a subspace of \(V\), the annihilator of this subspace is the collection of all linear functionals in \(V^*\) that map every vector in the subspace to zero. Its properties are useful in understanding how different spaces relate and interact. For instance, annihilators can reveal dependencies and help simplify complex systems by removing redundant components that add no meaningful dimension or structure to the analysis.