{"type":"rich","version":"1.0","title":"vector space basis definition","html":"<span class=\"wb-embed wb-embed-math\" data-latex=\"\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}\">\\forall \\vec v \\in V, \\vec v=\\operatorname{span}\\{\\vec u_1,\\dots,\\vec u_n\\}\n=\\left\\{\n\\sum_{i=1}^n \\lambda_i \\vec u_i\n\\;\\middle|\\;\n\\lambda_i \\in \\mathbb{F}\n\\right\\}</span><div class=\"wb-embed-note\"><p>And $\\vec u_1,\\dots,\\vec u_n$ are <a href=\"/wiki/Linearly_independent\" title=\"Linearly independent\">linearly independent</a>,\n</p><p>Then $B=\\{\\vec u_1,\\dots,\\vec u_n\\}$ is a basis of $V$\n</p><p>$n$ is known as the <b>dimension</b> of $V$\n</p></div><footer><cite class=\"wb-embed-author\">Rong Martin-Siebler ZHOU</cite></footer>","width":640,"height":320}