Thread: I need help on this Dimension theory HW (linear Alg)

Let V1,V2,.....Vk,V be vectors in a vector space V. and define W1 to be span(v1,V2,.....Vk) and W2 to be the Span(V1,V2.....Vk,V)

1) Find the necessary conditions on V such that dim(W1)=dim(W2)

2) State and prove a relationship involving dim(W1) and dim(W2), in the case where they dont equal each other.

Hello,

I can give you the solutions, and let you do the proof ! That's a good exercise.
(and this is also because I don't know how to do it )

Originally Posted by JCIR

Let V1,V2,.....Vk,V be vectors in a vector space V. and define W1 to be span(v1,V2,.....Vk) and W2 to be the Span(V1,V2.....Vk,V)

1) Find the necessary conditions on V such that dim(W1)=dim(W2)

V is a linear combination of V1,V2,.....,Vk

2) State and prove a relationship involving dim(W1) and dim(W2), in the case where they dont equal each other.

Hmm dim(W1)<dim(W2)

Originally Posted by JCIR

Let V1,V2,.....Vk,V be vectors in a vector space V. and define W1 to be span(v1,V2,.....Vk) and W2 to be the Span(V1,V2.....Vk,V)

1) Find the necessary conditions on V such that dim(W1)=dim(W2)

2) State and prove a relationship involving dim(W1) and dim(W2), in the case where they dont equal each other.

Note thus if and only if if and only if .

If they do not equal to eachother then we know that the dimension must be stricly less, so, .

Originally Posted by ThePerfectHacker

if and only if