# Gabriela Jeronimo, Juan Sabia y Susana Tesauri's Algebra Lineal PDF

By Gabriela Jeronimo, Juan Sabia y Susana Tesauri

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El conjunto de las matrices de n filas y m columnas con coeficientes en un cuerpo K es       a11 . . a1m   ..  . n×m ..  / aij ∈ K ∀ 1 ≤ i ≤ n, 1 ≤ j ≤ m . K =  .     an1 . . anm Para definir una matriz en K n×m basta especificar, para cada 1 ≤ i ≤ n y cada 1 ≤ j ≤ m, qu´e elemento de K se halla en el lugar ij (correspondiente a la intersecci´on de la fila i y la columna j) de la matriz. Ejemplo. Sean n, m ∈ N, y sean 1 ≤ k ≤ n, 1 ≤ l ≤ m. Se define la matriz E kl ∈ K n×m como 1 si i = k, j = l (E kl )ij = 0 si no Estas matrices se llaman las matrices can´ onicas de K n×m .

I) {(1, 1, 1, 1) , (0, 2, 1, 1)}, V = R4 , K = R ii) {X 3 − 2X + 1 , X 3 + 3X}, V = R3 [X], K = R iii) 1 i 1 1 , 0 1 i 1 , 0 1 2 1 , V = C2×2 , K = R y K = C Ejercicio 33. Extraer una base de S de cada uno de los siguientes sistemas de generadores. 5 Ejercicios 43 i) S = < (1, 1, 2) , (1, 3, 5) , (1, 1, 4) , (5, 1, 1) > ⊆ R3 , K = R ii) S = < X 2 + 2X + 1 , X 2 + 3X + 1 , X + 2 > ⊆ R[X], K = R iii) S = 1 1 1 1 0 i 1 1 , 0 i 0 0 , , 1 1 0 0 ⊆ C2×2 , K = R y K = C Ejercicio 34. i) Sea B = {f0 , f1 , f2 , .

Resolver los siguientes sistemas de ecuaciones lineales (K = R).   = 0  x1 + x2 − 2x3 + x4  x1 + x2 − 2x3 + x4 3x1 − 2x2 + x3 + 5x4 = 0 3x1 − 2x2 + x3 + 5x4 ii) i)   x1 − x2 + x3 + 2x4 = 0 x1 − x2 + x3 + 2x4   x1 + x2 + x3 − 2x4 + x5 x1 − 3x2 + x3 + x4 + x5 iii)  3x1 − 5x2 + 3x3 + 3x5 = 1 = 0 = 0   iv) x1 + x2 + x3 + x4 x1 + 3x2 + 2x3 + 4x4  2x1 + x3 − x4 = −2 = 3 = 2 = 2 = 0 = 6 ¿Cambia algo si K = Q? ¿Y si K = C? Ejercicio 12. i) Resolver los siguientes sistemas y comparar los conjuntos de soluciones (K = R).