Álgebra Linear c 4

u + v = (u₁, u₂, ..., uₖ) + (v₁, v₂, ..., vₖ) = (u₁ + v₁, u₂ + v₂, ..., uₖ + vₖ)\nau = (au₁, au₂, ..., auₖ) u + v = [u_11 u_12] + [v_11 v_12] = [u_11 + v_11 u_12 + v_12]\n [u_21 u_22] [v_21 v_22] [u_21 + v_21 u_22 + v_22]\n\na u = a [u_11 u_12] = [a u_11 a u_12]\n [u_21 u_22] [a u_21 a u_22]\n\ng = 0 [0 0]\n [0 0]\n\n0 + u = [ 0 0 ] + [u_11 u_12 ] = u\n [u_21 u_22 ]\n\n-u = [ -u_11 -u_12]\n [ -u_21 -u_22]\n\na + (-u) = [u_11 u_12 ] + [-u_11 -u_12] = [0 0]\n [u_21 u_22 ] [ -u_21 -u_22 ] \n\n1u = 1 [u_11 u_12 ] = [u_11 u_12 ] \n [u_21 u_22 ] [u_21 u_22 ] u + v = (2 + (-3), 4 + 5) = (-1, 9)\na u = 7u = (7 · 2, 0) = (14, 0)\n\n\nu + 1/u = u(1/(u)) = 1 (u = 0) 0u + 0u = (0 + 0)u [Axioma 8]\n= 0u [Propriedade do número 0]\n\n0u + [0u + (-0u)] = 0u + (-0u) [Axioma 3]\n0u + 0 = 0 [Axioma 5]\n0u = 0 [Axioma 4]\n\nu + (-1)u = 1u + (-1)u [Axioma 10]\n= (1 + (-1))u [Axioma 8]\n= 0u [Propriedade de números]\n= 0 [Parte (a) desse teorema] u + w = y + w\n(u + w) + (-w) = (y + w) + (-w)\nu + [w + (-w)] = y + [w + (-w)]\nu + 0 = y + 0\nu = v\n\nHipótese\nSomar -w a ambos lados\n\nu + 0 = v + 0\n\nu = v