linear_algebra-2-homework.ipynb

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    "%load_ext autoreload\n",
    "%autoreload 2\n",
    "\n",
    "%matplotlib inline"
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    "## Basic setup"
   ]
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    "Create anaconda environment\n",
    "<br>\n",
    "```bash\n",
    "conda create -n ml python=3.7.4 jupyter\n",
    "```\n",
    "Install fastai library\n",
    "<br>\n",
    "```bash\n",
    "conda install -c pytorch -c fastai fastai\n",
    "```"
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   "source": [
    "# Linear spaces"
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    "#### Prove that:\n",
    "For every linear function $f:V \\to U$ between vector spaces $V$ and $U$ on the real numbers, every sequence of vectors $v_1, v_2, \\dots v_m \\in V$ and every scalars $a_1, a_2, \\dots, a_m \\in \\mathbb{R}$:\n",
    "<br>\n",
    "$$f(a_1v_1 + a_2v_2 + \\dots + a_mv_m) = a_1f(v_1) + a_2f(v_2) + \\dots + a_mf(v_m)$$"
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    "Using first property of linear functions $f(x + y)  = f(x) + f(y)$:\n",
    "<br>\n",
    "$f(a_1v_1 + a_2v_2 + \\dots + a_mv_m) = f(a_1v_1) + f(a_2v_2) + \\dots + a_mf(a_mv_m)$\n",
    "<br>\n",
    "Using second property of linear functions: $f(cx) = cf(x)$, where $c$ is scalar:\n",
    "<br>\n",
    "$f(a_1v_1) + f(a_2v_2) + \\dots + a_mf(a_mv_m) = a_1f(v_1) + a_2f(v_2) + \\dots + a_mf(v_m)$\n",
    "<br>\n",
    "$\\blacksquare$"
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