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DEPARTAMENTO DE MATEMÁTICA APLICADA.
ESCUELA UNIVERSITARIA DE ARQUITECTURA TÉCMCA.
UNIVERSIDAD DE GRANADA
ÁLGEBRA LINEAL. SEGUNDO PARCIAL
Curso 200O~2001 - 8 de junio de 2001
GRUPO A
1. Posición relativa de una recta y un plano en el espacio afín tridimensional.
2. Sea f :R3- R3 la aplicación lineal dada por
f(x, y, z) = (x + y, -2x + z, 2z)
(a) Calcule la matriz asociada a f respecto de la base canónica de lft3
(b) Calcule el núcleo y la imagen de f. ¿Es inyectiva?, ¿es sobreyectiva? Justifique las respuestas.
(c) Calcule la matriz asociada a f respecto de la base B = {(1, 1,0), (0,1,1), (1, 0, 1)}.
3. Estudie para qué valores del parámetro real a es diagonalizable la matriz A=(fl ½ A)
Diagonalice la matriz A para = 2.
4. Considere los puntos del espacio afín euclldeo tridimensional
A=(1,1,0),B=(0,-1,2)yc=(3,0,-2).
(a) Calcule la distancia del punto A a la recta que pasa por B y por G.
(b) Determine el área del triángulo que determinan.
(c ) ~Determine un punto D de modo que el tetraedro de vértices A, B, C y D tenga volumen 2. ¿Es único este punto? En caso negativo, describa todos los puntos.
5 En el plano afín euclídeo considere la cónica de ecuación
4x2+ 9 y2- 24x - 36y +36 = O
(a) Clasifique la cónica.
(b) Calcule los elementos geométricos asociados a dicha cónica.
DURACIÓN: 2 horas 30 minutos.
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Enviado por: El remitente no desea revelar su nombre Idioma: castellanoPaís: España