1

Let $V$ and $W$ be finite dimensional vector spaces on a field $F$. Show that $\dim(V\oplus W) = \dim V +\dim W$.

My idea: let $\dim V=n$ and $\dim W=n$. So $\mathcal{A}=$ {${v_1 , v_2 ,... , v_n} $} and $\mathcal{B}=${${w_1, w_2 ,... ,w_m}$} this means $\forall v\in V$, $v=x_1 v_1 + ... +x_n v_n $ such that $x_1 , ... , x_n \in F$ . Similarly, $\forall w\in W$ , $w= y_1 w_1 + ... + y_m w_m$ such that $y_1 , ... , y_m \in F$ . I know as $V$ , $W$ have finite basis $V\oplus W$ has finite basis, I mean, $\forall (v,w)\in V\oplus W$ , $(v,w)=a_1 (v_1 ,w_1 ) + ... + a_p (v_p , w_p )$ such that $a_1 , ... , a_p \in F$ . Let $p$ is minimum between ${m,n}$ .Beside, $\forall (v,w)\in V\oplus W$ , $(v,w)=(v,0)+(0,w)$ . So we can say, $(v,w)= x_1 (v_1 ,0) + y_1 (0,w) +...+ x_p ( v_p , 0) + y_p (0 , w_p)$.

I want to know how I have to continue to get correct proof. Also, I want to write in mathematical way. I will appreciate any help.

Zeezee
  • 125

1 Answers1

4

If $B=\{v_1,\ldots,v_n\}$ is a basis for $V$ and $B'=\{w_1,\ldots,w_m\} $ is a basis for $W$, can you show $$B''=\{(v_i,0_W),(0_V,w_j):i=1,2,\ldots,n\;,\;j=1,\ldots,m\}$$ is a basis for $V\oplus W$? It should be clear it spans $V\oplus W$. So it remains to show it is linearly independent.

Pedro
  • 125,149
  • 19
  • 236
  • 403
  • first question is why $V\oplus W$ is spans? (what I the exact definition of span?) and the second is can I let $0_V\oplus W = ( 0_V , 0_W )$ ? – Zeezee Nov 25 '13 at 16:54