Question

Let \(V \stackrel{T}{\rightarrow} W \stackrel{S}{\rightarrow} V\) be linear transformations such that \(S T=1_{V}\). If \(\operatorname{dim} V=\operatorname{dim} W=n\), show that \(S=T^{-1}\) and \(T=S^{-1}\). [Hint: Exercise 7.3.13 and Theorem 7.3.3, Theorem 7.3.4, and Theorem 7.3.5.]

Short answer

S and T are inverses; S = T^{-1} and T = S^{-1}.

Step-by-step solution

Understand the Given Information

We have two vector spaces, \(V\) and \(W\), and two linear transformations \(T: V \rightarrow W\) and \(S: W \rightarrow V\). We know that \(ST = 1_V\), where \(1_V\) is the identity transformation on \(V\). Additionally, we are given that the dimensions of \(V\) and \(W\) are both \(n\).

Apply Theorem of Inverses

According to linear algebra theorems, if \(ST = 1_V\), then \(T\) must be injective and \(S\) must be surjective. Since \(T\) is injective and \(S\) is surjective and the dimensions of \(V\) and \(W\) are the same, we can conclude that both \(T\) and \(S\) are bijective.

Use Dimension Comparison

Since both transformations are bijective and the dimensions of \(V\) and \(W\) are equal, it follows that \(T\) and \(S\) serve as inverse transformations of one another on their respective spaces.

Conclude S and T are Inverses

Given the injective and surjective nature of \(T\) and \(S\), and the fact that \(ST = 1_V\), we can infer by comparing dimensions that both \(S = T^{-1}\) and \(T = S^{-1}\). This fits with the properties of inverse transformations in finite-dimensional spaces.

Key concepts

Linear Transformations

In linear algebra, transformations are fundamental in understanding how different vector spaces interact with each other. A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. For example, if we have two vector spaces, say \( V \) and \( W \), a linear transformation \( T: V \rightarrow W \) maps vectors from \( V \) to vectors in \( W \). This process respects the structure of the vector spaces, meaning that applying \( T \) to a sum of vectors is the same as applying \( T \) to each vector and then summing the results. Mathematically, this is described by the rule \( T(av + bw) = aT(v) + bT(w) \) for any vectors \( v \) and \( w \) in \( V \) and scalars \( a \) and \( b \).
Linear transformations simplify complex problems in mathematics, allowing us to work in different vector spaces while maintaining the structural integrity of the data at hand. They are powerful tools in computer graphics, engineering, and data science.

Vector Spaces

Vector spaces are a core component of linear algebra. A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars, forming what is known as a linear structure. The concept of a vector space is foundational because it provides a general framework in which we can discuss different kinds of operations and transformations.
To construct a vector space, we need:

• A set of vectors
• A field of scalars (often real or complex numbers)
• Defined operations of vector addition and scalar multiplication

These elements must satisfy several axioms like associativity, commutativity, identity, and distributive properties.
What's remarkable is that the same basic principles apply whether you're dealing with simple 2-dimensional spaces or more abstract and infinite-dimensional spaces. Understanding vector spaces allows for a unified approach to many areas of mathematics and its applications in science and engineering.

Inverse Transformations

Inverse transformations play a crucial role in analyzing the relationships between vector spaces. Given two transformations \( T: V \rightarrow W \) and \( S: W \rightarrow V \), if the composition of these transformations \( ST = 1_V \) acts as the identity on vector space \( V \), we can deduce that each transformation has an inverse. This means \( S \) is the inverse of \( T \), denoted as \( S = T^{-1} \), and vice versa, \( T = S^{-1} \).
In the context of finite-dimensional vector spaces, a transformation's invertibility is linked to its bijectivity. For a transformation to have an inverse, it must be both injective (one-to-one) and surjective (onto). When these conditions are met, the transformation applies a unique output for each input, and every element of the target space has a pre-image in the domain. This property is vital for solving systems of equations and transforming data in various applications.

Dimensionality

Dimensionality refers to the number of independent vectors in a vector space, often thought of as the number of dimensions in that space. If a vector space \( V \) has a basis consisting of \( n \) vectors, then the dimension of \( V \) is \( n \). The basis is essentially a minimal set of vectors that span the entire space, meaning any vector in the space can be represented as a linear combination of basis vectors.
In linear transformations, dimensionality helps determine properties like injectivity and surjectivity. When transforming between vector spaces \( V \) and \( W \) of equal dimension, if a transformation is either injective or surjective, it is indeed bijective. This implies invertibility, as seen in the provided exercise where the dimensions of \( V \) and \( W \) are the same. Understanding dimensionality is key to solving numerous problems in linear algebra, from determining solution families to ensuring transformations are reversible.