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# HW - Math 601 HOMEWORK 1 1 Decide which of the following...

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Math 601 HOMEWORK 1 1. Decide which of the following sets are subspaces in R 3 , and for each subspace ﬁnd a basis, its dimension and describe the type of geometrical ﬁgure it is. U 1 = { ( x 1 ,x 2 ,x 3 ) | x 1 = 2 t, x 2 = - t, x 3 = 5 t, t R } U 2 = { ( x 1 ,x 2 ,x 3 ) | x 1 = 2 t + 1 , x 2 = - t, x 3 = 5 t, t R } U 3 = { ( x 1 ,x 2 ,x 3 ) | x 1 = 2 t 2 , x 2 = - t, x 3 = 5 t, t R } U 4 = { ( x 1 ,x 2 ,x 3 ) | x 1 = 2 t, x 2 = - t, x 3 = 5 t, t > 0 } U 5 = Sp ( x,y,z ) with x = (1 , 1 , 0) , y = ( - 2 , - 2 , 0) , z = (0 , 1 , 1) 2. Show that the intersection U V of two subspaces U, V is also a subspace. 3. Show that the sum U + V of two subspaces U, V is also a subspace. 4. Consider the vector space of polynomials of degree at most n , with coeﬃcients in R : P n = { p ( t ) = a 0 + a 1 t + a 2 t 2 + ... + a n t n | a j R } Show that the monomials 1 , t,t 2 ,... t n form a basis for P n . Show that
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