Suppose v1 v2 v3 v4 spans V Prove that the list v1 v2 v2 v

Suppose v₁, v₂, v₃, v₄ spans V. Prove that the list v₁ - v₂, v₂ - v₃, v₃ - v₄, v₄ also spans V.

Let, x be a vector in span of v1,v2,v3,v4

So, x=av1+bv2+cv3+dv4=av1-av2+av2+bv2+cv3+dv4

=a(v1-v2)+(b+a)v2-(b+a)v3+(b+a)v3+cv3+dv4

=a(v1-v2)+(b+a)(v2-v3)+(b+a+c)v3-(b+a+c)v4+(b+a+c)v4+dv4

=a(v1-v2)+(b+a)(v2-v3)+(b+a+c)(v3-v4)+(a+b+c+d)v4

Hence, x is in span of: v1-v2,v2-v3,v3-v4 and v4

Now let y be in span of:v1-v2,v2-v3,v3-v4 and v4

So,

y=a(v1-v2)+b(v2-v3)+c(v3-v4)+dv4

y=av1+(b-a)v2+(c-b)v3+(d-c)v4

Hence, y is in span of :v1,v2,v3,v4