Question

Consider a linear operator \(\mathbf{T}\) on a finite-dimensional vector space \(V\). (a) Show that there exists a polynomial \(p\) such that \(p(\mathbf{T})=\mathbf{0}\). Hint: Take a basis \(B=\left\\{\left|a_{i}\right\rangle\right\\}_{i=1}^{N}\) and consider the vectors \(\left\\{\mathbf{T}^{k}\left|a_{1}\right\rangle\right\\}_{k=0}^{M}\) for large enough \(M\) and conclude that there exists a polynomial \(p_{1}(\mathbf{T})\) such that \(p_{1}(\mathbf{T})\left|a_{1}\right\rangle=0\). Do the same for \(\left|a_{2}\right\rangle\), etc. Now take the product of all such polynomials. (b) From (a) conclude that for large enough \(n, \mathbf{T}^{n}\) can be written as a linear combination of smaller powers of \(\mathbf{T}\). (c) Now conclude that any infinite series in \(\mathbf{T}\) collapses to a polynomial in \(\mathbf{T}\).

Short answer

For any linear operator T on a finite-dimensional vector space V, there is a polynomial p such that p(T)=0. For sufficiently large powers, the operator can be expressed as a linear combination of smaller powers of T. Any infinite series in T collapses to a polynomial in T.

Step-by-step solution

Using Linear Dependency in Vectors

Given a finite-dimensional vector space V and a linear operator T on it, the aim is to find a polynomial p such that p(T)=0. In any finite-dimensional vector space, a set with more elements than the dimension of the space is linearly dependent. Hence, for a large enough M, the set of vectors {T^k|a_1>} for k=0,1,...,M is linearly dependent and therefore, by the basic properties of linear dependencies, there exist coefficients c_k such that the sum c_k*T^k|a_1>=0. This sum is equivalent to a polynomial p1(T) with the property that p1(T)|a_1>=0.

Constructing the Polynomial for Each Basis Vector

This step can be repeated for all basis vectors |a_i>. For each basis vector |a_i>, there exists a polynomial p_i(T) such that p_i(T)|a_i>=0. Therefore, for any vector |v> in the space V, which can be written as the linear combination of the basis vectors |a_i>, there exists a polynomial p(T) which gives p(T)|v>=0, where p(T) is just the product of all the polynomials p_i(T).

Demonstrating That T^n Can Be Written as a Linear Combination of Smaller Powers of T

The polynomial p(T) can be written in the form of T^n plus lower order terms, as we can simply combine powers to get the polynomial in this form. However, since p(T)=0, this means that T^n can be written as the negative of the term involving smaller powers of T, which is a linear combination of smaller powers of T.

Collapse of Infinite Series

Considering the series expansion of any infinite series in powers of T, there will be terms with sufficiently large powers, and these can be written as a linear combination of smaller powers. By continually applying this process, the series will collapse to a polynomial in T. This is feasible because once the power reaches the order of the polynomial, the process can kick in and the series will begin to collapse.

Key concepts

Finite-Dimensional Vector Space

A finite-dimensional vector space is an essential structure in linear algebra where the number of vectors forming its basis is finite. This characteristic enables us to perform various analyses using linear operators. Let's break down what this means:

• A vector space is a collection of vectors that can be stretched, shrunk, added, and combined in multiple ways.
• In finite-dimensional spaces, you can describe every vector using linear combinations of a finite number of basis vectors.

The finite nature of these spaces simplifies many operations as it dictates constraints on things like linear dependence, which we'll cover next. It also allows us to apply the concept of polynomials in linear operators, enabling us to express and manipulate operators concisely.

Linear Dependence

Linear dependence is a fundamental concept in vector spaces that occurs when a vector is expressible as a combination of other vectors. For example, consider a set of vectors in a finite-dimensional space:

• If any vector within this set can be written as a linear combination of the others, the set is considered linearly dependent.
• This characteristic allows us to reduce complexities when dealing with large vector sets.
• In our exercise, if you have more vectors in a set than the dimension of the space, they must be linearly dependent.

This property is extremely useful, especially in finding polynomials that annihilate vectors (make them zero when operated upon), which leads us deeper into polynomial representation in linear operators.

Polynomial Representation

Polynomial representation in the context of linear operators allows us to express operators in terms of polynomial functions. This leads to significant simplifications:

• In the exercise, if a linear operator \( \mathbf{T} \) has a polynomial representation such that it nullifies all basis vectors, it essentially nullifies the entire space.
• This is achieved by identifying polynomials for each basis vector, multiplying them together, and applying them to the linear operator.
• This polynomial representation facilitates the expression of \( \mathbf{T}^n \) as a linear combination of its preceding powers, which is crucial for simplifying calculations.

By expressing linear operators polynomially, we gain a powerful tool for handling otherwise complex transformations in vector spaces.

Infinite Series Collapse

The idea of an infinite series collapsing to a polynomial is an important application of linear operators on finite-dimensional spaces. Here’s how this works:

• In an infinite series involving the operator \( \mathbf{T} \), any series term with a sufficiently large power can be written as a combination of smaller powers.
• This is due to the existence of a polynomial \( p(T) \) that zeroes out after a certain degree, meaning higher powers can be substituted with these lower degree combinations.
• When you apply this logic throughout the series, it simplifies and reduces to a finite polynomial form.

The collapse of an infinite series to a polynomial showcases the elegance of tidy mathematical properties ensuring computational tasks remain manageable.