{ "cells": [ { "cell_type": "code", "execution_count": 42, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "text/html": [ " " ], "text/plain": [ "" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.core.display import HTML as Center\n", "\n", "Center(\"\"\" \"\"\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Linear algebra - crash course\n", "\n", "We will start by recalling from linear algebra the definition of a vector space and a linear map. These notion will be crucial in the definition and interpretation of group representations that we will introduce in the following lectures.\n", "\n", "## Vector spaces\n", "\n", "```{admonition} Definition (Vector space)\n", ":class: definition\n", "\n", "A *vector space* over a field $\\mathbb{F}$ ($\\mathbb{R}$ or $\\mathbb{C}$), is a set $V$ together with two operations: vector addition and multiplication by scalar, that satisfy the following axioms:\n", "- vector addition is commutative: $v+u=u+v$ for all $u,v\\in V$\n", "- vector addition is associative: $(v+u)+w=v+(u+w)$ for all $v,u,w\\in V$\n", "- vector addition has an identity element (zero vector 0) satisfying $v+0=0+v=v$ for all $v\\in V$\n", "- vector addition has inverses: for all $v\\in V$ there exists $u\\in V$ such that $v+u=u+v=0$ \n", "- $\\lambda(\\mu v)=(\\lambda\\mu)v$ for all $\\lambda,\\mu\\in\\mathbb{F}$ and $v\\in V$\n", "- $\\lambda(v+u)=\\lambda v+\\lambda u$ for all $\\lambda\\in\\mathbb{F}$ and $v,u\\in V$\n", "- $(\\lambda+\\mu) v=\\lambda v+\\mu v$ for all $\\lambda,\\mu\\in\\mathbb{F}$ and $v\\in V$\n", "```\n", "\n", "$n$-dimensional Euclidean spaces are the most familiar examples of vector spaces. However, there are a lot more spaces and operations that satisfy these axioms. \n", "\n", "```{admonition} Examples (Vector space)\n", ":class: example\n", "\n", "- $\\mathbb{F}=\\mathbb{R}$, $V=\\mathbb{R}^n$ is a vector space over real numbers\n", "- $\\mathbb{F}=\\mathbb{C}$, $V=\\mathbb{C}^n$ is a vector space over complex numbers\n", "- the set $M_{n\\times m}(\\mathbb{R})$ of all $n\\times m$ matrices is a vector space\n", "\n", "```\n", "\n", "We will be dealing mostly with finite dimensional vector spaces in these lectures. In such a case, we can introduce the notion of a basis of a vector space. We start by defining what we mean by linearly independent vectors:\n", "\n", "```{admonition} Definition (Linearly independent vectors)\n", ":class: definition\n", "\n", "Let $V$ be an $n$-dimensional vector space and let $v_1,v_2,\\ldots,v_k\\in V$ be vectors. Then we say that these vectors are linearly independent if the vector equation\n", "\n", "$$\n", "\\alpha_1 v_1+\\alpha_2 v_2+\\ldots \\alpha_k v_k=0\n", "$$\n", "\n", "has only the trivial solution $\\alpha_1=\\alpha_2=\\ldots=\\alpha_k=0$.\n", "```\n", "\n", "```{admonition} Example (Linearly independent vectors)\n", ":class: example\n", "\n", "Let us take $V=\\mathbb{R}^2$ and take three vectors $v_1=\\begin{pmatrix}2\\\\3\\end{pmatrix}$, $v_2=\\begin{pmatrix}-1\\\\1\\end{pmatrix}$ and $v_3=\\begin{pmatrix}2\\\\-2\\end{pmatrix}$. One can show that vectors $v_1$ and $v_2$ are linearly independent since\n", "\n", "$$\n", "\\alpha_1 v_1+\\alpha_2 v_2=\\alpha_1 \\begin{pmatrix}2\\\\3\\end{pmatrix}+\\alpha_2 \\begin{pmatrix}-1\\\\1\\end{pmatrix}=\\begin{pmatrix}2 \\alpha_1-\\alpha_2\\\\3\\alpha_1+\\alpha_2\\end{pmatrix}\n", "$$\n", "\n", "and to have $\\alpha_1 v_1+\\alpha_2 v_2=0$ we need to solve two equations:\n", "\n", "$$\n", "2 \\alpha_1-\\alpha_2=0,\\qquad 3\\alpha_1+\\alpha_2=0\n", "$$\n", "\n", "The only solution is $\\alpha_1=\\alpha_2=0$.\n", "\n", "On the other hand, vectors $v_2$ and $v_3$ are not linearly independent (therefore they are linearly dependent), since for example $2v_2+v_3=0$.\n", "```\n", "\n", "Finally, we can define a basis of a vector space as\n", "\n", "```{admonition} Definition (Basis of vector space)\n", ":class: definition\n", "\n", "Let $V$ be an $n$-dimensional vector space. Any set of $n$ linearly independent vectors $v_1,v_2,\\ldots , v_n$ in $V$ is called the basis of $V$.\n", "```\n", "\n", "Importantly, given a basis $v_1,v_2,\\ldots,v_n$ of a vector space $V$, it is possible to expand any other vector $v\\in V$ as a linear combination of these basis vectors. Such linear combination is unique.\n", "\n", "```{admonition} Example (Basis of vector space)\n", ":class: example\n", "\n", "Let us take $V=\\mathbb{R}^2$. We showed that the vectors $v_1=\\begin{pmatrix}2\\\\3\\end{pmatrix}$ and $v_2=\\begin{pmatrix}-1\\\\1\\end{pmatrix}$ are linearly independent. Therefore, $v_1$ and $v_2$ form a basis of $\\mathbb{R}^2$. Any other vector in $\\mathbb{R}^2$ can be written as a linear combination of these two vectors. For example, let us take $v=\\begin{pmatrix}5\\\\7\\end{pmatrix}$. Then\n", "\n", "$$\n", "v=\\frac{12}{5}v_1-\\frac{1}{5}v_2\n", "$$\n", "\n", "```\n", "\n", "## Linear subspaces\n", "\n", "When discussing representation theory, we will often consider a vector space of some dimension $n$, and will be interested in its linear subspaces. If $V$ is a vector space over a field $K$ and if $W$ is a subset of $V$, then $W$ is a *linear subspace* of $V$ if under the operations of $V$, $W$ is a vector space over $K$. This can be illustrated by the following example\n", "\n", "```{admonition} Example (Linear subspace)\n", ":class: example\n", "\n", "Let $V=\\mathbb{R}^2$, and let us define a subset of $V$: $W=\\{(x,y)\\in \\mathbb{R}^2:-3x+2y=0\\}$. Then $W$ is a vector space and therefore a linear subspace of $V$. Since the elements of $W$ are just the 2-dimensional vectors, then the axioms of vector spaces are easily satisfied. The only thing that we need to check is whether $W$ is closed under vector addition and multiplication by a scalar. Let us first notice that all element of $W$ can be written as \n", "\n", "$$\n", "W=\\{(x,\\frac{3}{2}x):x\\in \\mathbb{R}\\}\n", "$$\n", "\n", "We start by checking that $W$ is closed under vector addition. Let us take $w_1,w_2\\in W$. Then we can write them as $w_1=(x_1,\\frac{3}{2}x_1)$ and $w_2=(x_2,\\frac{3}{2}x_2)$ for some $x_1,x_2\\in \\mathbb{R}$. Then\n", "\n", "$$\n", "w_1+w_2=(x_1,\\frac{3}{2}x_1)+(x_2,\\frac{3}{2}x_2)=(x_1+x_2,\\frac{3}{2}(x_1+x_2))=(x_3,\\frac{3}{2}x_3)\n", "$$\n", "\n", "for $x_3=x_1+x_2$. Therefore $w_1+w_2\\in W$.\n", "\n", "Similarly, we can check that $W$ is closed under the multiplication by a scalar. Let us take $w\\in W$ and $\\lambda\\in \\mathbb{R}$. Then $w=(x,\\frac{3}{2}x)$ for some $x\\in\\mathbb{R}$ and\n", "\n", "$$\n", "\\lambda w=\\lambda (x,\\frac{3}{2}x)=(\\lambda x,\\frac{3}{2}\\lambda x)=(x' ,\\frac{3}{2}x')\n", "$$\n", "\n", "for $x'=\\lambda x\\in \\mathbb{R}$. Therefore, $\\lambda w\\in W$.\n", "\n", "It is also easy to find a basis of $W$. Since $\\dim(W)=1$ then it is sufficient to find a single linearly independent vector. As an example we can take $w=(2,3)$. \n", "\n", "Moreover, the linear subspace $W$ can be easily visualized on a real plane (that is $\\mathbb{R}^2$). It is just the straight line in the picture below\n", "\n", "```" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": "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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "\n", "point_1 = [4,6]\n", "point_2 = [-4,-6]\n", "\n", "\n", "ax = plt.gca()\n", "ax.set_aspect('equal', 'box')\n", "ax.grid()\n", "ax.set_xlim(-7,7)\n", "ax.set_ylim(-7,7)\n", "\n", "x_values = [[point_1[0]],[point_2[0]]]\n", "y_values = [[point_1[1]],[point_2[1]]]\n", "\n", "plt.plot(x_values, y_values, 'red')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Direct sum of linear subspaces\n", "\n", "One additional notion from linear algebra that will be very useful to us is a direct sum of vector spaces. \n", "\n", "```{admonition} Definition (Direct sum of vector spaces)\n", ":class: definition\n", "\n", "Let $V$ be a vector space over the field $\\mathbb{F}$. Let $W_1$ and $W_2$ be two linear subspaces of $V$. We define the **direct sum** $W_1\\oplus W_2$ of $W_1$ and $W_2$ as\n", "\n", "$$\n", "W_1\\oplus W_2=\\{w_1+w_2:w_1\\in W_1,w_2\\in W_2\\}\n", "$$\n", "```\n", "\n", "## Linear maps\n", "\n", "Finally, we recall the notion of a linear map between two vector spaces:\n", "\n", "```{admonition} Definition (Linear map)\n", ":class: definition\n", "\n", "Let $U$ and $V$ be vector spaces over the field $\\mathbb{F}$. Then $f:U\\to V$ is called a linear map if\n", "- $f(u+v)=f(u)+f(v)$ for all $u,v\\in U$\n", "- $f(\\lambda u)=\\lambda f(u)$ for all $\\lambda\\in \\mathbb{F}$ and $u\\in U$\n", "\n", "```\n", "\n", "```{admonition} Examples (Linear map)\n", ":class: example\n", "\n", "- The identity map $id_U:U\\to U$ is a linear map\n", "- For an $n\\times m$ real matrix $A$, we can define a map: $T_A:\\mathbb{R}^n\\to\\mathbb{R}^m$ as $T_A(v)=A\\cdot v$. Then $T_A$ is a linear map.\n", "\n", "```" ] } ], "metadata": { "celltoolbar": "Edit Metadata", "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.7" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": {}, "version_major": 2, "version_minor": 0 } } }, "nbformat": 4, "nbformat_minor": 4 }