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

1. ## 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.

2. 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)

3. 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 $\displaystyle W_1\subseteq W_2$ thus $\displaystyle \mbox{dim}(W_1) = \mbox{dim}(W_2)$ if and only if $\displaystyle W_1 = W_2$ if and only if $\displaystyle V \in \text{spam}(V_1,...,V_k)$.

If they do not equal to eachother then we know that the dimension must be stricly less, so, $\displaystyle \text{dim}(W_1) < \text{dim}(W_2)$.

4. Originally Posted by ThePerfectHacker
if and only if $\displaystyle V \in \text{spa{\color{red}m}}(V_1,...,V_k)$