Homework SourseSuppose v1 v2 and v3 are vectors in R n such that v3 v1 v2
Suppose v1, v2, and v3 are vectors in R n such that v3 = v1 + v2. Show that Span{v1, v2, v3} = Span{v1, v2} by proving the following statements. (a) Prove that if y Span{v1, v2, v3}, then y Span{v1, v2}. (b) Prove that if x Span{v1, v2}, then x Span{v1, v2, v3}
Solution
(a). Let y be an arbitrary vector in Span {v1, v2, v3}. Then y is a linear combination of the vectors v1, v2, v3. Let y = av1+bv2+cv3, where a,b,c are arbitrary scalars. Since v3 = v1 + v2, we have y =av1+bv2+c(v1 + v2) = (a+c)v1+(b+c)v2. Thus, y is a linear combination of the vectors v1, v2 . Hence y Span{v1, v2}.
(b) Let x be an arbitrary vector in Span{v1, v2}. Then x is a linear combination of the vectors v1, v2. Let x = av1+bv2, where a,b are arbitrary scalars. Then x = av1+[(b-a)v2+av2]= a(v1 + v2)+ (b-a)v2 = av3 + (b-a)v2. Thus, x is a linear combination of the vectors v2, v3 . Hence x Span {v1, v2,v3}.
Therefore Span{v1, v2, v3} Span{v1, v2} and Span{v1, v2} {v1, v2, v3} so that Span{v1, v2, v3} = Span{v1, v2}.
