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Fin rendering in LinearAlgebra
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remilvus authored Jul 29, 2022
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8 changes: 4 additions & 4 deletions tutorials/LinearAlgebra/LinearAlgebra.ipynb
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"\n",
"Another, equivalent definition highlights what makes this an interesting property. For any matrices $B$ and $C$ of compatible sizes:\n",
"\n",
"$$A^{-1}(AB) = A(A^{-1}B) = B \\\\\n",
"(CA)A^{-1} = (CA^{-1})A = C$$\n",
"$$A^{-1}(AB) = A(A^{-1}B) = B$$\n",
"$$(CA)A^{-1} = (CA^{-1})A = C$$\n",
"\n",
"A square matrix has a property called the **determinant**, with the determinant of matrix $A$ being written as $|A|$. A matrix is invertible if and only if its determinant isn't equal to $0$.\n",
"\n",
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" A_{n-1,0} \\cdot \\color{blue} {\\begin{bmatrix}B_{0,0} & \\dotsb & B_{0,l-1} \\\\ \\vdots & \\ddots & \\vdots \\\\ B_{k-1,0} & \\dotsb & B_{k-1,l-1} \\end{bmatrix}} & \\dotsb &\n",
" A_{n-1,m-1} \\cdot \\color{red} {\\begin{bmatrix}B_{0,0} & \\dotsb & B_{0,l-1} \\\\ \\vdots & \\ddots & \\vdots \\\\ B_{k-1,0} & \\dotsb & B_{k-1,l-1} \\end{bmatrix}}\n",
"\\end{bmatrix}\n",
"= \\\\\n",
"=\n",
"=$$\n",
"$$=\n",
"\\begin{bmatrix}\n",
" A_{0,0} \\cdot \\color{red} {B_{0,0}} & \\dotsb & A_{0,0} \\cdot \\color{red} {B_{0,l-1}} & \\dotsb & A_{0,m-1} \\cdot \\color{blue} {B_{0,0}} & \\dotsb & A_{0,m-1} \\cdot \\color{blue} {B_{0,l-1}} \\\\\n",
" \\vdots & \\ddots & \\vdots & \\dotsb & \\vdots & \\ddots & \\vdots \\\\\n",
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