Suppose that the vectors v1 v2 vk vk 1 span the vector spa

Suppose that the vectors v_1, v_2, ..., v_k, v_k + 1 span the vector space V and that v_k + 1 is a linear combination of v_1, v_2,..., v_k. Show that the vectors v_1, v_2,..., v_k span V.

Solution

suppose that the vectors v1,v2 ........vk,vk+1 span the vecotr space V   

then V = a1v1+a2v2+a3v3+......................+akvk+ak+1 vk+1 --------------->1

here a1,a2.......................ak+1 are any constants

and vk+1 is linear combination of vectors v1,v2,............vk

vk+1 = b1v1 +b2v2+.........................+bkvk

here b1,b2.......................................bk are constants

now plug vk+1 value in equation1

V = a1v1+a2v2+a3v3+......................+akvk+ak+1 (b1v1 +b2v2+.........................+bkvk)

V = a1v1+a2v2+a3v3+......................+akvk+ + ak+1 b1v1 + ak+1b2 v2 +..........+ak+1bkvk

V = v1(a1+ak+1b1) +v2(a2+ak+1b2) +...........................+(ak+ak+1bk)vk

here a1+ak+1b1 will also a constant

therfore v1,v2,v3...................vk span V