there is

v1,..,vk,u,w vectors on space V

the group {v1,..,vk,u} is independant

and

find the dimention of sp{v1..vk,u,w}

?

if {v1,..,vk,u} is independant then its span dimention is k+1

and span so {v1,..,vk} independat to and dim sp{v1,..,vk}=k

the dimention of sp{v1..vk,u,w} is

dim(sp{v1..vk,u,w})=dim(sp{v1..vk}and sp{u,w}})=dim(sp{v1..vk}+sp{u,w})=dim(sp{v1..vk})+ dim(sp{u,w})-dim(sp{v1..vk} intersect sp{u,w}})

if v1..vk,u is independant then v1,..,vk is independant too so dim (sp{v1..vk})=k

now regardingthe other two members i dont know