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Chapter 8: Problem 9

Suppose \(T \in \mathcal{L}(V)\) and \(m\) is a nonnegative integer such that range \(T^{m}=\) range \(T^{m+1}\). Prove that range \(T^{k}=\) range \(T^{m}\) for all \(k > m\).

Short Answer

Expert verified
We demonstrated that range \(T^k\) is a subset of range \(T^m\) and range \(T^m\) is a subset of range \(T^k\) for all \(k > m\), which implies that range \(T^k = T^m\) for all \(k > m\).

Step by step solution

01

Prove that range \(T^k\) is a subset of range \(T^m\) for all \(k > m\)#

Let's consider an element \(y \in\) range \(T^k\), for some \(k > m\). This means that there exists \(x \in V\) such that \(y = T^k(x)\). Now, since range \(T^m = T^{m+1}\), by definition, there exists \(z \in V\) such that \(T^m(z) = T^{m+1}(z)\). Since \(T^m = T^{m+1}\), we can apply the transformation \(T^{-1}\) (inverse) to both sides of the equation: \[ T^{-1}(T^m(z)) = T^{-1}(T^{m+1}(z)) \] Simplifying, we get \(T^{m-1}(z) = T^m(z)\). Now, returning to \(y = T^k(x)\), since \(k > m\), we can rewrite it as: \[ y = T^m(T^{k-m}(x)) \] Using the fact that \(T^{m-1}(z) = T^m(z)\), we can rewrite this expression as: \[ y = T^{m-1}(T^{k-m+1}(x)) \] Since \(k > m\), \(k-m+1 > 1\), so that \(T^{k-m+1}(x) \in V\). Therefore, \(y \in\) range \(T^m\). This implies that the range \(T^k\) is a subset of range \(T^m\) for all \(k > m\).
02

Prove that range \(T^m\) is a subset of range \(T^k\) for all \(k > m\)#

Now, let's consider an element \(y \in\) range \(T^m\). This means that there exists \(x \in V\) such that \(y = T^m(x)\). Since range \(T^m = T^{m+1}\), we have \(y = T^{m+1}(x) = T^m(T(x))\). Now, let \(z = T(x)\). Then, \(y = T^m(z)\), where \(z \in V\), since \(T(x) = z\) is just applying the linear transformation \(T\) to \(x\). It implies that \(y = T^m(z) = T^{k}(T^{m-k}(z))\), where \(k > m\). Recall that \(k > m\), meaning that \(m-k < 0\). However, since \(T^m = T^{m+1}\), the transformation is invertible (i.e., it has an inverse). Therefore, we can write \(T^{-(k-m)}(z) = T^{-1}(T^{k-m}(z))\). With this, we can rewrite the expression as: \[ y = T^k(T^{-(k-m)}(z)) \] This means that \(y \in\) range \(T^k\) since \(T^{-(k-m)}(z) \in V\). Therefore, the range \(T^m\) is a subset of range \(T^k\) for all \(k > m\).
03

Conclusion#

Since range \(T^k\) is a subset of range \(T^m\) and range \(T^m\) is a subset of range \(T^k\) for all \(k > m\), we can conclude that range \(T^k = T^m\) for all \(k > m\). This completes the proof.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Transformation
A linear transformation is a fundamental concept in linear algebra. It is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means if you have two vectors, say \( u \) and \( v \), in your vector space along with a scalar \( c \), then a linear transformation \( T \) from one vector space to another will satisfy two main properties:
  • Additivity: \( T(u + v) = T(u) + T(v) \).
  • Homogeneity: \( T(cu) = cT(u) \).
These properties ensure that the structure of the vector space is maintained under the transformation. Linear transformations can be represented by matrices when dealing with finite-dimensional vector spaces, which makes them incredibly useful in both practical applications and theoretical work. Understanding linear transformations provides the basis for many areas in mathematics, including understanding the behavior of operators like the one discussed in the exercise.
Range of an Operator
The range of an operator, particularly a linear operator, is an essential concept that describes the set of all possible outputs 鈥� or images 鈥� of a linear transformation. If \( T: V \rightarrow W \) is a linear operator where \( V \) and \( W \) are vector spaces, then the range of \( T \), denoted as \( ext{range}(T) \), is defined as:
  • \( ext{range}(T) = \{ T(v) \mid v \, \in \, V \} \).
This definition implies that any vector in the range of \( T \) can be expressed as \( T(v) \) for some vector \( v \) in the domain \( V \). An important aspect of understanding a linear operator's range involves recognizing when changes occur in this range, such as in the given exercise, where consistency in the range across higher powers \( T^m = T^{m+1} \) indicates a stabilization. This stabilization often facilitates proving certain properties about linear operators, like in understanding invariant subspaces or projecting operations.
Subset in Vector Spaces
Understanding subsets within vector spaces is crucial when discussing linear transformations and their properties. A subset is simply a collection of vectors that altogether form part of a larger vector space. Let's consider two vector spaces: \( U \) and \( V \); \( U \) is a subset of \( V \) if every vector in \( U \) is also a vector in \( V \). Subsets are not necessarily vector spaces on their own unless they satisfy the conditions for being a vector space (closure under addition and scalar multiplication, etc.). When dealing with operators, like the linear operator \( T \) from the exercise, the range and the domain, which are themselves subsets, gain significance. To prove that one range is a subset of another, we show that applying the transformation \( T \) or its power, maps vectors into a particular subset of another vector space, implying containment. As described in the solution steps of the exercise, the demonstration that one range is a subset of another is a preliminary step in proving equality between ranges when they're impacted by the transformation's repeated application.

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Most popular questions from this chapter

Suppose \(a_{0}, \ldots, a_{n-1} \in \mathrm{C} .\) Find the minimal and characteristic polynomials of the operator on \(\mathrm{C}^{n}\) whose matrix (with respect to the standard basis) is $$\left[\begin{array}{cccc} 0 & & & & -a_{0} \\ 1 & 0 & & & -a_{1} \\ & 1 & \ddots & & -a_{2} \\ & & \ddots & & \vdots \\ & & & 0 & -a_{n-2} \\ & & & 1 & -a_{n-1} \end{array}\right]$$

Define \(T \in \mathcal{L}\left(\mathbf{C}^{2}\right)\) by \\[T(w, z)=(-z, w).\\] Find all generalized eigenvectors of \(T\).

Give an example of an operator on \(\mathrm{C}^{4}\) whose characteristic polynomial equals \(z(z-1)^{2}(z-3)\) and whose minimal polynomial equals \(z(z-1)(z-3)\).

Give an example of an operator on \(\mathbf{C}^{3}\) whose minimal polynomial equals \(z^{2}\).

Give an example of an operator on \(\mathrm{C}^{4}\) whose characteristic polynomial equals \((z-7)^{2}(z-8)^{2}\).
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