la_06_vector_space_basis.ipynb

{"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"provenance":[],"authorship_tag":"ABX9TyNoPNeBSWQKPoVc+8zHpMcW"},"kernelspec":{"name":"python3","display_name":"Python 3"},"language_info":{"name":"python"}},"cells":[{"cell_type":"markdown","source":["#### Concept of vector space"],"metadata":{"id":"FJTiDzNH80z0"}},{"cell_type":"markdown","source":["A `collection` of vectors that can be added together and scaled by scalars, `while still staying within the collection`\n","\n","For example, n-dimensional space $\\mathbf{R}^n$ is a vector space"],"metadata":{"id":"GgPVDsAX9ACT"}},{"cell_type":"markdown","source":["#### Concept of subspace"],"metadata":{"id":"uFtr-d5b9mUf"}},{"cell_type":"markdown","source":["Intuitively, a `subspace` of a vector space is a `subset` of a vector space\n","\n","The following three are all `subspaces` of $\\mathbf{R}^n$\n","\n","$$\\begin{align*}\n","V_1&=\\mathbf{R}^n \\\\\n","V_2&=\\{0\\}, 0 \\in \\mathbf{R}^n \\\\\n","V_3&=\\text{span}(v_1, v_2, \\cdots, v_k), v_i \\in \\mathbf{R}^n\n","\\end{align*}$$"],"metadata":{"id":"FFUmogHNXZlo"}},{"cell_type":"markdown","source":["#### Independent set of vectors"],"metadata":{"id":"h6eZHgB-h3xr"}},{"cell_type":"markdown","source":["A set of vectors $v_1, v_2, \\cdots, v_k$ is `independent` if\n","\n","$$\\alpha_1v_1+\\alpha_2v_2+\\cdots +\\alpha_kv_k=0 \\Longrightarrow \\alpha_1=\\alpha_2=\\cdots = \\alpha_k = 0$$"],"metadata":{"id":"JuipZqvjh7w4"}},{"cell_type":"markdown","source":["As a result, we have the following\n","\n","$$\\alpha_1v_1+\\alpha_2v_2+\\cdots +\\alpha_kv_k=\\beta_1v_1+\\beta_2v_2+\\cdots +\\beta_kv_k \\Longrightarrow \\alpha_i=\\beta_i, i=1, 2,\\cdots, k$$\n","\n","and `no` set of nonzero coefficients $\\alpha_i$ can satisfy\n","\n","$$v_j=\\alpha_1v_1+\\cdots+\\alpha_{j-1}v_{j-1}+\\alpha_{j+1}v_{j+1}+\\cdots+\\alpha_kv_k$$\n","\n","That is, no $v_j$ can be expressed as a linear combination of the remaining vectors (can be seen via contradiction)"],"metadata":{"id":"wEEITOhXSgF8"}},{"cell_type":"markdown","source":["#### Basis"],"metadata":{"id":"RzSDlfgPUCzC"}},{"cell_type":"markdown","source":["$v_1, v_2, \\cdots, v_k$ is a set of `basis vectors` for vector space $V$ if\n","\n","* $V=\\text{span}(v_1, v_2, \\cdots, v_k)$\n","* $v_1, v_2, \\cdots, v_k$ form an `independent` set"],"metadata":{"id":"e7kWq1sOUEzo"}},{"cell_type":"markdown","source":["For every $v\\in V$, it can be `uniquely` expressed using basis vectors\n","\n","$$v=\\alpha_1v_1+\\alpha_2v_2+\\cdots +\\alpha_kv_k$$"],"metadata":{"id":"cpKXENAxVb6b"}},{"cell_type":"markdown","source":["The most common basis for $\\mathbf{R}^n$ is $e_1, e_2, \\cdots, e_n$, which is also `orthonormal`, that is\n","\n","$$e_i^Te_j = \\left\\{\\begin{array}{cl}0 &i\\neq j \\\\1 &i=j \\end{array}\\right.$$"],"metadata":{"id":"r6vwID_RV_Vr"}},{"cell_type":"markdown","source":["For a given vector space $V$, the `number of vectors` in any basis is the same\n","\n","The number of vectors in any basis is the `dimension` of the corresponding vector space $\\dim V$\n","\n","If $v_1, v_2, \\cdots, v_k$ form a set of `basis vectors` for $V$, then\n","\n","$$\\dim V=k$$\n","\n","As $v_i\\in \\mathbf{R}^n$, $V$ is a subspace of $\\mathbf{R}^n$"],"metadata":{"id":"nrwF9_QmYB_N"}},{"cell_type":"code","source":[],"metadata":{"id":"sYRONfQG9fHc"},"execution_count":null,"outputs":[]}]}