Homework SourseLet V be a vector space and let v1 v2 elementof V Prove that
Let V be a vector space, and let v_1, v_2 elementof V. Prove that span v_1, v_2 = span v_1, v_2, v_1 + v_2.
Solution
Let v be a vector in span {v1,v2,v1+v2}. Then v is a linear combination of v1,v2 and v1+v2. Let v = av1 +bv2 +c(v1+v2), where a,b,c are scalars. Then v = (a+c)v1 +(b+c)v2. This means that v span {v1,v2} so that span {v1,v2,v1+v2} span {v1,v2}. Now, let u span {v1,v2}. Then u is a linear combination of v1,v2 . Let u = av1+bv2. Then u = (a-c)v1+(b-c)v2 +c (v1+v2). This implies that u span {v1,v2, v1+v2 } so that span {v1,v2} span {v1,v2,v1+v2}. Hence, span {v1,v2} =span {v1,v2,v1+v2}.
