3.3.1

Let $$v_1 = (1,2,-1,0), v_2 = (4,8,-4,3), v_3 = (0,1,3,4), v_4 = (2,5,1,4) $$, and let W be the subspace of $$\mathbb{R}^4$$ spanned by these four vectors. Observe that $$2v_1 + v_3 = v_4$$, so that $$W = span\{v_1,v_2,v_3\}$$. We claim that these three vectors are linearly independent, and thus form a basis for W. Suppose there exist some real numbers $$c_1,c_2,c_3$$ such that $$c_1v_1+c_2v_2+c_3v_3 = 0$$, then $$(c_1+4c_2,2c_1+8c_2+c_3,-c_1-4c_2+3c_3,3c_2+4c_3) = (0,0,0,0)$$. Then, $$c_1+4c_2 = 0$$ and $$-c_1-4c_2+3c_3 = 0$$, so that $$c_3 = 0$$, thus using this result in $$3c_2 + 4c_3 = 0$$, we find $$c_2 = 0$$, which finally implies $$c_1 = 0$$. Thus the vectors $$v_1,v_2,v_3$$ are linearly independent and hence form a basis for W.