Question

Is every isomorphism \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\) given by an invertible matrix \(U\) such that \(T(X)=U X\) for all \(X\) in \(\mathbf{M}_{22}\) ? Prove your answer.

Short answer

Yes, every isomorphism is given by an invertible matrix multiplication.

Step-by-step solution

Understand the Problem

We are given a map \( T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22} \) and need to determine if every isomorphism is represented by multiplication with an invertible matrix \( U \). \( \mathbf{M}_{22} \) represents the space of 2x2 matrices.

Define Isomorphism in Linear Algebra

An isomorphism is a linear bijection. For a map \( T \) to be an isomorphism, it must be both linear and invertible. This means \( T(A + B) = T(A) + T(B) \) and \( T(cA) = cT(A) \) for all \( A, B \in \mathbf{M}_{22} \) and scalar \( c \).

Consider Representations of Linear Maps

In finite dimensions, every linear transformation from one vector space to itself can be represented as matrix multiplication by a fixed matrix. Here, it means there exists some matrix \( U \) such that \( T(X) = UX \) for any \( X \in \mathbf{M}_{22} \).

Check for Invertibility

We want \( U \) to be invertible for \( T \) to be an isomorphism. If \( U \) is invertible, then \( T \) is bijective, as every nonzero vector has a unique image and preimage.

Proof by Construction

Prove that any isomorphism \( T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22} \) is given by multiplication with an invertible matrix. Construct any map \( T \) and show that it can be expressed as \( T(X) = UX \) with an invertible \( U \).

Verify Completeness

Having shown that any isomorphism can be expressed as \( UX \) with \( U \) invertible, we conclude this representation is complete for all isomorphisms on \( \mathbf{M}_{22} \). Thus, any isomorphism is given by multiplication with an invertible matrix \( U \).

Key concepts

Linear Transformation

In linear algebra, a linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. Imagine you have a set of vectors, which we can think of as arrows, and you're transforming them into another set of vectors. A linear transformation ensures that when you add two vectors or multiply a vector by a scalar, these actions are maintained after the transformation.
For example, if you have vectors \( A \) and \( B \), a linear transformation \( T \) would satisfy \( T(A + B) = T(A) + T(B) \). Similarly, if \( c \) is a scalar, then \( T(cA) = cT(A) \). These properties are crucial, as they ensure the linear structure of the space is preserved.
Linear transformations can be represented by matrices and are foundational for understanding more complex operations in linear algebra, such as rotations and scaling in graphics and data transformations.

Invertible Matrix

An invertible matrix, often called a non-singular or full-rank matrix, is a square matrix that has an inverse. The property of being invertible is important because it indicates that the matrix can uniquely map vectors from one space to another and back again.
If a matrix \( U \) is invertible, it means there exists another matrix \( U^{-1} \) such that \( UU^{-1} = I \), where \( I \) is the identity matrix. The identity matrix is special because it acts like the number 1 in matrix multiplication, leaving any matrix it multiplies unchanged.
In the context of the exercise, if the matrix \( U \) in the transformation \( T(X) = UX \) is invertible, then the transformation \( T \) can be reversed. This reversibility is essential for isomorphisms because it ensures each element in the domain maps to a unique element in the codomain and vice versa.

Matrix Representation

Matrix representation is a powerful way to express linear transformations. Essentially, it allows us to use matrices to perform operations on vectors within the vector space. When we say a transformation \( T \) can be represented by a matrix \( U \), it means that applying \( T \) to a vector \( X \) is the same as performing the matrix multiplication \( UX \).
This concept is central to simplifying computations in linear algebra, especially in higher dimensions. Instead of working directly with abstract transformations, we use matrices to facilitate calculations, predict outcomes, and analyze changes in vector spaces.
Real-life applications include computer graphics, where transformations like rotations and translations are easily performed using matrices. Thus, matrix representation is not just a theoretical tool but also a practical way to handle complex transformations efficiently.

Bijective Mapping

Bijective mapping is a function between two sets that is both injective (one-to-one) and surjective (onto). In simpler terms, every element in the first set is paired with a unique element in the second set, and every element in the second set is covered.
In terms of linear transformations and matrices, a transformation is bijective if, and only if, its matrix representation is invertible. This is because an invertible matrix ensures that each vector in the original space has a unique image and a unique pre-image in the transformed space.
Why does this matter? Bijective mappings are crucial for isomorphisms in linear algebra. They ensure that transformations are reversible, meaning we can "go back" to the original configuration without losing any information or configuration. This reversible nature is vital in fields such as cryptography and data compression, where reversing a transformation seamlessly is essential.