1. Indicate if the following functions ?: ?^2? ?^2 are linear transformations, justifying your answer. 1.1. ?(?, ?) = (1 + ?, ?). 1.2. ?(?, ?) = (?, ?) 1.3 ?(?, ?) = (?^2, ?) 1.4 ?(?, ?) = (????, ?) 1.5. ?(?, ?) = (? - ?, 0). 1.6. ?(?, ?) = (|?|, 0).

2. Suppose ?: ? ? ? and ?: ? ? ? are linear transformations. Show that the composition transformation (? ? ?: ? ? ? ? ? (? ? ?)(?) = ?(?(?)); is a linear transformation.

3. Let ?: ? ? ? be a linear transformation, and suppose that the vectors ?1, ?2, ? , ?n ? ? have the property that their images ?(?1), ?(?2), ? , ?(?n) are linearly independent. Prove that ?1, ?2, ? , ?n are also linearly independent.

4. Let ?: ?^2? ?^2 be the linear transformation that rotates a vector with respect to the Z-axis and with an angle ?: ?(?, ?, ?) = (????? - ?????, ????? + ?????, ?). 4.1. Find ???(?). 4.2. Find ??(?).

(The image is the original questions in spanish.)

1. Indique si las siguientes funciones 7: 13 - R3 son transformaciones lineales, justificando su respuesta. 1.1. (x, y) = (1+x,y) 1.4. T(x, y) = (senx, y ) 1.2. T(x, y) = (V,x) 1.5. 7(x, y) = (x - y, 0) 1.3. T(x, y) = (x],y) 1.6. T(x, y) = (Ixlo) 2. Suponga que F: V - U y G:U - W son transformaciones lineales. Muestre que la transformation composition (G . F: V - Wa (G . F)(x) = G(F()) es una transformation lineal. 3. Sea 7:1 - I una transformation lineal, y suponga que los vectores X1, X2, "., Xx EV tienen la propiedad que sus imagenes 7(), (x2), ....7( ) son linealmente independientes. Pruebe que X1, x2, ...,xx son tambien linealmente independentes. 4. Sea F: R - R la transformation lineal que rota un vector con respecto al eje Z y con un angulo 0: F(x,y,z) = (xcos0 - ysend, xsend + ycose, z). 4.1. Encuentro Ker (F) 4.2. Encuentro Im (F)