Question

In each case, show that \(T\) is an isomorphism by defining \(T^{-1}\) explicitly. a. \(T: \mathbf{P}_{n} \rightarrow \mathbf{P}_{n}\) is given by \(T[p(x)]=p(x+1)\). b. \(T: \mathbf{M}_{n n} \rightarrow \mathbf{M}_{n n}\) is given by \(T(A)=U A\) where \(U\) is invertible in \(\mathbf{M}_{n n}\)

Short answer

Both transformations are isomorphisms with inverses \( T^{-1}[p(x)] = p(x-1) \) for part (a) and \( T^{-1}(B) = U^{-1}B \) for part (b).

Step-by-step solution

Understand Isomorphism in Linear Algebra

An isomorphism is a bijective linear transformation between two vector spaces. To show that a transformation \( T \) is an isomorphism, we need to prove that it is both injective (one-to-one) and surjective (onto). Additionally, we need to define its inverse \( T^{-1} \) explicitly.

Define and Verify the Transformation for Part (a)

For part (a), the transformation \( T : \mathbf{P}_n \to \mathbf{P}_n \) is given by \( T[p(x)] = p(x+1) \). We must show this function is bijective by defining its inverse. Let's define \( T^{-1} : \mathbf{P}_n \to \mathbf{P}_n \) by \( T^{-1}[q(x)] = q(x-1) \). \[T^{-1}[T[p(x)]] = T^{-1}[p(x+1)] = p((x+1)-1) = p(x)\]and\[T[T^{-1}[q(x)]] = T[q(x-1)] = q((x-1)+1) = q(x).\] Thus, \( T^{-1} \) is indeed the inverse of \( T \), establishing that \( T \) is bijective.

Define and Verify the Transformation for Part (b)

For part (b), the transformation \( T: \mathbf{M}_{nn} \to \mathbf{M}_{nn} \) is defined by \( T(A) = UA \), where \( U \) is an invertible matrix. Define \( T^{-1}: \mathbf{M}_{nn} \to \mathbf{M}_{nn} \) by \( T^{-1}(B) = U^{-1}B \). \[T^{-1}[T(A)] = T^{-1}[UA] = U^{-1}(UA) = (U^{-1}U)A = IA = A\]and\[T[T^{-1}(B)] = T[U^{-1}B] = U(U^{-1}B) = (UU^{-1})B = IB = B.\] Therefore, \( T^{-1} \) is the inverse of \( T \), confirming that \( T \) is bijective.

Key concepts

Isomorphism

In linear algebra, an isomorphism is a special kind of transformation. It bridges two vector spaces through a bijective linear transformation. This implies two key characteristics:

• Injective (One-to-One): Every element in the first vector space maps to a unique element in the second.
• Surjective (Onto): Every element in the target vector space is accounted for by the mapping.

When a transformation is both injective and surjective, it can be "inverted." This means there exists an inverse function that can map elements back to their original form. Demonstrating that a transformation is an isomorphism involves showing it meets these criteria and explicitly defining this inverse function.

Inverse Function

The concept of an inverse function is essential in isomorphisms. For a function to have an inverse means that each input is linked to a unique output, and vice versa. In other words, if you perform a transformation on data and then apply the inverse transformation, you should retrieve the original data.
When you're asked to explicitly define an inverse function, you're essentially asked to "undo" the transformation. Consider the examples from the exercise:

• For polynomial transformation, the function that shifts a polynomial by one unit can be undone by shifting back one unit.
• For matrix transformations, multiplying by an invertible matrix can be undone by multiplying by its inverse.

The inverse functions, therefore, prove that both transformations can be reversed correctly.

Bijective

A function is bijective if it is both injective and surjective. Bijectivity is fundamental to proving that a transformation is an isomorphism.
Bijection means:

• Injective: The function never maps two different elements of the domain to the same point in the codomain. Thus, each input has a unique output.
• Surjective: The function covers the entire codomain, meaning every possible output is used at least once.

In the context of the exercise, establishing bijectivity involves showing that each transformation grabs every bit of the codomain, exhibiting a perfect pairing between the domain and codomain.

Invertible Matrix

An invertible matrix is a key player in transformations that are isomorphisms. A matrix is said to be invertible if there exists another matrix that, when multiplied with it, results in the identity matrix. This property is crucial for defining inverse functions in transformations.
Key properties of invertible matrices include:

• The determinant of an invertible matrix is non-zero.
• Multiplying an invertible matrix by its inverse returns an identity matrix.

In part (b) of the exercise, the transformation relies on multiplying by an invertible matrix. The ability to invert the matrix makes it possible to apply the transformation and then apply its inverse to return to the start. This provides a concrete example of how invertible matrices work in practice to define bijective transformations.