--- Commentary: $\textbf{Exercise 20.}$ Suppose $p_0, p_1, \ldots, p_m$ are polynomials in $\mathcal{P}_m(\mathbb{F})$ such that $p_k(2) = 0$ for each $k \in {0, \ldots, m}$. Prove that $p_0, p_1, \ldots, p_m$ is not linearly independent in $\mathcal{P}_m(\mathbb{F})$. $\textbf{Solution 20.}$ Because $p_k(2) = 0$ for each $k$, the constant polynomial 1 is not in $\operatorname{span}(p_0, \ldots, p_m)$. Hence if the list $p_0, p_1, \ldots, p_m$ was linearly independent, then the list $p_0, p_1, \ldots, p_m, 1$ would also be linearly independent. But this is impossible in $\mathcal{P}_m(\mathbb{F})$ because this list has length $m + 2$, which is larger than the length of the spanning list $1, z, \ldots, z^m$. $\textit{Commentary:}$ This exercise presents a condition that forces a list of polynomials to be linearly dependent. The condition is that all the polynomials in the list evaluate to zero at a specific point (in this case, at $x = 2$). This implies that the constant polynomial 1 is not in the span of these polynomials, because 1 does not evaluate to zero at 2. The proof then argues by contradiction. If the list of polynomials were linearly independent, then adding 1 to the list would yield a linearly independent list of length $m + 2$. But this is impossible in the space $\mathcal{P}_m(\mathbb{F})$ of polynomials of degree at most $m$, because this space has dimension $m + 1$ (as witnessed by the spanning list $1, z, \ldots, z^m$). This exercise provides practice in reasoning about linear independence and spans in polynomial spaces. It also illustrates how evaluating polynomials at specific points can provide information about their linear independence. $\textit{Examples:}$ 1. If $p_0, p_1, \ldots, p_m \in \mathcal{P}_m(\mathbb{R})$ all have a zero at $x = 0$, then they are not linearly independent. This is because 1 is not in their span. 2. If $p_0, p_1, \ldots, p_m \in \mathcal{P}_m(\mathbb{C})$ all have a zero at $z = i$, then they are not linearly independent. Again, 1 is not in their span. 3. In $\mathcal{P}_3(\mathbb{Q})$, the polynomials $x - 1$, $x^2 - 1$, $x^3 - 1$ are not linearly independent, because they all have a zero at $x = 1$. 4. In $\mathcal{P}_4(\mathbb{F}_5)$, the polynomials $x - 2$, $x^2 - 4$, $x^3 + 3x$, $x^4 + x^2 + 1$ are not linearly independent, because they all have a zero at $x = 2$ (in $\mathbb{F}_5$). These examples demonstrate the same principle in various polynomial spaces, over different fields and with different evaluation points. In each case, the fact that all polynomials in the list evaluate to zero at a specific point implies that they are linearly dependent. This is a useful tool for showing linear dependence in polynomial spaces without having to explicitly find a non-trivial linear combination that equals zero. In conclusion, the exercises in Section 2C provide a comprehensive exploration of the concept of dimension in vector spaces. They cover topics such as finding bases for subspaces, extending bases, direct sum decompositions, classifying subspaces by dimension, and infinite-dimensional spaces. The exercises also showcase a variety of techniques for proving results about dimension, such as using the spanning list theorem, the linear dependence lemma, and the exchange lemma. The commentaries and examples help to clarify the concepts and illustrate their applications in different vector spaces over various fields. Overall, mastering these exercises would provide a strong foundation in the theory of dimension and its role in linear algebra. --- ## Quantum mechanics Infinite dimensional Hilbert spaces play a fundamental role in quantum theory. In fact, the state space of a quantum system is typically modeled as an infinite dimensional Hilbert space. This is a profound and far-reaching connection that underlies much of modern physics. In quantum mechanics, the state of a system is represented by a vector in a Hilbert space, usually denoted as $|\psi\rangle$. This state vector encodes all the information about the system. Observables, such as position, momentum, and energy, are represented by linear operators acting on this Hilbert space. One of the key features of quantum mechanics is the principle of superposition, which states that if $|\psi_1\rangle$ and $|\psi_2\rangle$ are possible states of a system, then so is any linear combination $a|\psi_1\rangle + b|\psi_2\rangle$ (where $a$ and $b$ are complex numbers). This is directly related to the linear structure of Hilbert spaces. Moreover, the inner product in a Hilbert space is used to calculate probabilities in quantum mechanics. Specifically, if $|\psi\rangle$ is the state of a system, and $|\phi\rangle$ is the state corresponding to a particular measurement outcome, then the probability of that outcome is given by $|\langle\phi|\psi\rangle|^2$ (where $\langle\phi|\psi\rangle$ is the inner product of $|\phi\rangle$ and $|\psi\rangle$). The fact that these Hilbert spaces are typically infinite dimensional is crucial. It allows for the description of systems with continuous observables (like position and momentum), and it's necessary for modeling systems with an infinite number of possible states (like a harmonic oscillator). Now, let's speculate about potential relationships with large language models and generative models in machine learning. Language models, like GPT-3, are typically based on neural networks with a vast number of parameters. In a sense, the space of possible states of these networks is also very high dimensional, although not infinite dimensional. One could imagine representing the state of a language model as a vector in a high-dimensional space, where each dimension corresponds to a particular parameter or feature. The output of the model, then, could be thought of as a function of this state vector. In this view, the process of training a language model could be seen as moving the state vector to a region of the space that corresponds to realistic, coherent language. The model's ability to generate diverse outputs could be related to the size and structure of this region. Similarly, generative models in machine learning, such as GANs and VAEs, also operate in high-dimensional spaces. The latent space of a GAN, for example, is a high-dimensional space that the generator maps to the space of images (or other types of data). Drawing a loose analogy with quantum mechanics, one could think of the latent space as a kind of "state space" for the generated data, and the generator as an operator that maps from this state space to the observable data space. Of course, these are just speculative analogies, and there are significant differences between these machine learning models and quantum systems. For one, the spaces involved in machine learning are not typically equipped with an inner product, which is a crucial feature of Hilbert spaces. Moreover, the "dynamics" of these machine learning models (i.e., how they generate outputs) are determined by the learned parameters of the model, rather than by a fixed set of physical laws. Nonetheless, exploring these analogies could potentially lead to interesting cross-fertilization between quantum theory and machine learning. Ideas from quantum information theory, for example, have already been applied to certain machine learning problems. In conclusion, while infinite dimensional Hilbert spaces are fundamentally woven into the fabric of quantum mechanics, their relevance to current machine learning models is more speculative. However, as machine learning models become increasingly complex and high-dimensional, it's not inconceivable that deeper connections to the mathematical structures of quantum theory could emerge. This is an exciting area for future research and exploration. --- ## Transformers and algorithmic improvements Transformers have indeed revolutionized the field of natural language processing (NLP) and have become the backbone of many state-of-the-art large language models. Let's discuss how transformers have changed the landscape of language modeling and then speculate on the potential impact of improvements in matrix multiplication. Transformers and Large Language Models: Transformers, introduced in the seminal paper "Attention Is All You Need" by Vaswani et al. (2017), have fundamentally changed the architecture of language models. Prior to transformers, recurrent neural networks (RNNs) and long short-term memory (LSTM) networks were the dominant architectures for language modeling. However, these models struggled with capturing long-range dependencies and suffered from vanishing gradient problems. Transformers addressed these limitations by introducing the concept of self-attention. In a transformer, each word in a sequence attends to every other word, allowing the model to directly capture long-range dependencies. This is achieved through the computation of attention weights, which determine how much each word should attend to every other word. Moreover, transformers introduced the concept of positional encoding, which allows the model to incorporate information about the position of each word in the sequence. This is crucial for capturing word order and syntactic structure. The use of transformers has led to significant improvements in language modeling performance. Models like BERT (Bidirectional Encoder Representations from Transformers) and GPT (Generative Pre-trained Transformer) have achieved state-of-the-art results on a wide range of NLP tasks, including sentiment analysis, named entity recognition, and question answering. Transformers have also enabled the creation of extremely large language models, such as GPT-3, which has 175 billion parameters. These models exhibit impressive language generation capabilities and can perform tasks like translation, summarization, and even code generation. Impact of Matrix Multiplication Improvements: The computation in transformers heavily relies on matrix multiplications. The attention mechanism, in particular, involves computing the dot product between query, key, and value matrices. As the size of these matrices grows with the size of the model, the computational cost of these matrix multiplications becomes a significant bottleneck. Improvements in matrix multiplication algorithms or hardware could have a significant impact on the efficiency of training and deploying large language models. Let's speculate on a few potential developments: 1. Faster Matrix Multiplication Algorithms: The development of faster matrix multiplication algorithms could reduce the time complexity of the attention computation. For example, the current best algorithm for matrix multiplication has a time complexity of O(n^2.37), which is better than the naive O(n^3) algorithm. If even faster algorithms were discovered, it could lead to significant speedups in transformer computations. 2. Specialized Hardware: The development of specialized hardware for matrix multiplications could greatly accelerate transformer computations. We've already seen the impact of GPUs and TPUs (Tensor Processing Units) on deep learning. The development of hardware specifically optimized for the types of matrix multiplications used in transformers could lead to further speedups. 3. Distributed Computing: Improvements in distributed computing frameworks and algorithms for matrix multiplication could allow for the training of even larger transformer models. By parallelizing the computation across multiple machines or even multiple data centers, it may be possible to train models with trillions of parameters. 4. Approximation Techniques: The development of approximation techniques for matrix multiplication, such as low-rank approximations or sparsity-based methods, could reduce the computational cost of transformer models. These techniques could potentially allow for the creation of models with similar performance but lower computational requirements. In conclusion, transformers have already profoundly changed the field of language modeling, enabling the creation of models with unprecedented performance and scale. Improvements in matrix multiplication, whether through algorithmic advances, hardware developments, distributed computing, or approximation techniques, could further push the boundaries of what is possible with these models. As the computational efficiency of transformers improves, we can expect to see even larger and more powerful language models in the future.