Question

Let \(T: V \rightarrow W\) be a linear transformation. a. If \(T\) is one-to-one and \(T R=T R_{1}\) for transformations \(R\) and \(R_{1}: U \rightarrow V\), show that \(R=R_{1}\). b. If \(T\) is onto and \(S T=S_{1} T\) for transformations \(S\) and \(S_{1}: W \rightarrow U,\) show that \(S=S_{1}\).

Short answer

R = R_1 and S = S_1.

Step-by-step solution

Understand One-to-One Transformation

A linear transformation \(T: V \to W\) is one-to-one if different elements in \(V\) map to different elements in \(W\). More formally, for \(x, y \in V\), if \(T(x) = T(y)\) implies \(x = y\). This property will be used to show that \(R = R_1\).

Apply One-to-One Property

Given that \(T R = T R_1\), apply the one-to-one property of \(T\). For any \(u \in U\), \(T(R(u)) = T(R_1(u))\). Since \(T\) is one-to-one, this implies that \(R(u) = R_1(u)\) for every \(u \in U\).

Conclude for One-to-One Case

Since \(R(u) = R_1(u)\) for all \(u \in U\), it follows that the transformations \(R\) and \(R_1\) are equal. Thus, \(R = R_1\).

Understand Onto Transformation

A linear transformation \(T: V \to W\) is onto if every element in \(W\) is the image of some element in \(V\). This means that for every \(w \in W\), there exists \(v \in V\) such that \(T(v) = w\). This property will be essential to show that \(S = S_1\).

Apply Onto Property

Given that \(S T = S_1 T\), apply the onto property of \(T\). For any \(v \in V\), \(S(T(v)) = S_1(T(v))\). Since \(T\) is onto, every \(w \in W\) has a pre-image \(v \in V\), implying \(S(w) = S_1(w)\) for every \(w \in W\).

Conclude for Onto Case

Since \(S(w) = S_1(w)\) for all \(w \in W\), it follows that the transformations \(S\) and \(S_1\) are equal. Thus, \(S = S_1\).

Key concepts

One-to-One Mappings

A one-to-one mapping, also known as an injective mapping, is a fascinating concept in linear algebra. This occurs when a linear transformation uniquely maps every distinct input in the vector space \(V\) to a unique output in another vector space \(W\). Simply put, if \(T: V \to W\) is one-to-one, no two different elements in \(V\) will be mapped to the same element in \(W\). This can be formally stated as, if \(T(x) = T(y)\), then it must follow that \(x = y\).

If you imagine this as giving out personalized gifts, where each person in \(V\) receives a unique gift in \(W\), the gifts cannot be the same for two different people. This important property means that the transformation is reversible, in the sense that you can always trace back to find the original element from the vector space \(V\) given an element in \(W\).

In practice, if you happen to show that transformations \(R\) and \(R_1\) from another space \(U\) to \(V\) satisfy \(T R = T R_1\) and \(T\) is one-to-one, this must imply \(R = R_1\). Just like discovering that two identical gifts imply they came from the exact same person in \(V\).

Onto Transformations

Understanding onto transformations is an essential step in grasping linear transformations fully. An onto transformation, or surjective map, means that every element of the vector space \(W\) is the image of some element in \(V\). In other words, \(T: V \to W\) is onto if for every \(w \in W\), there is at least one \(v \in V\) such that \(T(v) = w\).

This scenario can be likened to making sure that every position or seat in \(W\) gets assigned to someone from \(V\). No seat remains unoccupied; each has at least one person matched to it. Therefore, the full range of \(W\) can be filled by elements from \(V\).

When it comes to equal transformations in this context, if you have two transformations \(S\) and \(S_1\) from \(W\) to another space \(U\), and if \(ST = S_1T\), it forces \(S\) and \(S_1\) to be identical when \(T\) is onto. It's as if agreeing to accept the same person in a seat results in the same kind of arrangement all around \(U\). This property is crucial for ensuring that every image can be translated back to an appropriate source, creating a complete linkage between \(V\) and \(W\).

Vector Spaces

Vector spaces form the backdrop for exploring linear transformations. A vector space \(V\) is an entirely structured set that consists not just of elements (or vectors), but supports operations like addition and scalar multiplication, following specific rules.

Imagine this as a wide play area where vectors can lie anywhere as long as they adhere to certain rules like closure under addition and scalar multiplication, existence of a zero vector, and more. These structures make vector spaces rich and varied fields, characterized by their dimensionality and basis.

These spaces have properties that allow transformations to be defined. And through those transformations, they build relationships with other vector spaces, like \(W\), in the journey of mapping one space to another through transformations such as \(T: V \rightarrow W\). Each operation within and between these spaces must preserve the rigorous rules that define the space. Linear transformations, such as those discussed in one-to-one and onto scenarios, derive their power and application from these structured, rule-bound environments.

Vector spaces are the realm where transformations showcase their true essence, demonstrating how such systems can alter, preserve, and relate diverse elements, helping us solve complex equations and problems in linear algebra.