45582e5f59
- 68 exercises from UBA FCE chapters 1-3 - Step-by-step solutions with KaTeX rendering - Theory panels (26 topics) expandable per exercise - Matrix builder (2x2/3x3/4x4) with 7 operations - System solver (Gauss, Gauss-Jordan, Cramer, Rouché-Frobenius) - Glassmorphism UI with dark mode - Canvas particle background - ARIA accessibility (keyboard nav, screen reader) - Zero build step - open index.html directly
42 lines
2.8 KiB
JSON
42 lines
2.8 KiB
JSON
{
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"matrices-theory": {
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"title": "Definición de Matriz",
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"content": "Una matriz es un arreglo rectangular de números dispuestos en filas y columnas. Una matriz de orden $m \\times n$ tiene $m$ filas y $n$ columnas. Notación: $A = [a_{ij}]$ donde $i$ indica la fila y $j$ la columna.",
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"relatedExercises": ["cap02-01", "cap02-02"]
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},
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"matrix-ops": {
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"title": "Operaciones con Matrices",
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"content": "Suma: $(A+B)_{ij} = a_{ij} + b_{ij}$ (mismo orden). Multiplicación por escalar: $(cA)_{ij} = c \\cdot a_{ij}$. Producto: $(AB)_{ij} = \\sum_{k} a_{ik} \\cdot b_{kj}$ (columnas de A = filas de B).",
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"relatedExercises": ["cap02-03", "cap02-05", "cap02-06"]
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},
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"matrix-types": {
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"title": "Matrices Especiales",
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"content": "Cuadrada: $n \\times n$. Identidad $I_n$: $1$ en diagonal, $0$ en el resto. Diagonal: $a_{ij}=0$ para $i \\neq j$. Triangular superior: $a_{ij}=0$ para $i > j$. Traspuesta $A^T$: $(A^T)_{ij} = a_{ji}$.",
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"relatedExercises": ["cap02-07", "cap02-08", "cap02-09"]
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},
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"transpose-symmetry": {
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"title": "Traspuesta y Simetría",
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"content": "Traspuesta: $(A^T)^T = A$, $(AB)^T = B^T A^T$. Simétrica: $A = A^T \\Leftrightarrow a_{ij} = a_{ji}$. Antisimétrica: $A = -A^T \\Leftrightarrow a_{ii}=0$. Toda matriz: $A = \\frac{A+A^T}{2} + \\frac{A-A^T}{2}$ (simétrica + antisimétrica).",
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"relatedExercises": ["cap02-10", "cap02-11"]
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},
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"determinants": {
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"title": "Determinantes",
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"content": "Orden 2: $|\\begin{matrix} a & b \\\\ c & d \\end{matrix}| = ad - bc$. Orden 3 (Sarrus): suma de 3 diagonales principales menos 3 diagonales secundarias. Cofactor: $C_{ij} = (-1)^{i+j} M_{ij}$ (menor con signo).",
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"relatedExercises": ["cap02-12", "cap02-13", "cap02-14"]
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},
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"determinant-props": {
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"title": "Propiedades de los Determinantes",
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"content": "$|A^T| = |A|$. Si se intercambian dos filas, cambia signo. Fila de ceros $\\Rightarrow |A|=0$. Filas proporcionales $\\Rightarrow |A|=0$. $F_i \\rightarrow F_i + cF_j$ no cambia $|A|$. $|AB| = |A| \\cdot |B|$. $|cA| = c^n |A|$ para $n \\times n$.",
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"relatedExercises": ["cap02-17", "cap02-18", "cap02-19"]
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},
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"inverse-matrix": {
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"title": "Matriz Inversa",
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"content": "$A$ invertible $\\Leftrightarrow |A| \\neq 0 \\Leftrightarrow \\text{rg}(A) = n$. $A^{-1} = \\frac{1}{|A|} \\text{adj}(A)$. Para $2 \\times 2$: $A^{-1} = \\frac{1}{ad-bc} \\begin{matrix} d & -b \\\\ -c & a \\end{matrix}$.",
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"relatedExercises": ["cap02-20", "cap02-21", "cap02-22", "cap02-24", "cap02-25"]
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},
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"rank": {
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"title": "Rango de una Matriz",
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"content": "Rango = orden del menor no nulo mayor. Equivale al número de filas no nulas en forma escalonada. Operaciones elementales (permutar, multiplicar por $c \\neq 0$, $F_i + cF_j$) no cambian el rango.",
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"relatedExercises": ["cap02-26", "cap02-27", "cap02-28", "cap02-29"]
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}
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} |