# Thread: Prove that W_1 and W_2 are independent: proof needs checking

1. ## Prove that W_1 and W_2 are independent: proof needs checking

I have a good deal of work done for this, but could use a second opinion (and could use some info on whether I'm missing anything).

Suppose that $V$ and $W$ are vector spaces over a field $F$.

Let $W_1$ and $W_2$ be subspaces of a finite-dimensional vector space $V$.
Prove that $W_1$ and $W_2$ are independent if and only if $W_1\cap W_2=\{0\}$.

Here is my work so far (well, it's how my Professor showed us during his office hours, so he probably made it incomplete).

$\forall w_1\in W_1$ and $\forall w_2\in W_2$ such that $w_1+w_2=0$, then $w_1=w_2=0$
If we suppose that $W_1,W_2$ are independent, this implies that $W_1\cap W_2=\{0\}$.
Suppose that $w\in W_1\cap W_2$,
Clearly, $w+(-w)=0$ (We can let $w_1=w$ and $w_2=-w$)
$\Rightarrow w=0$

Using the above, we now need to prove that $W_1\cap W_2=\{0\}\Rightarrow W_1,W_2$ are independent.
Suppose $w_1\in W_1$ and $w_2\in W_2$ and that $w_1+w_2=0$.
$\Rightarrow w_1=-w_2$ OR $w_2=-w_1$
$\Rightarrow w_1\in W_2$ OR $w_2\in W_1$

These imply that $w_1=w_2=0$, and that they are independent.

As such, $W_1$ and $W_2$ are independent.

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Tell me if my answer needs any clean-up or reordering.

2. Originally Posted by Runty
I have a good deal of work done for this, but could use a second opinion (and could use some info on whether I'm missing anything).

Suppose that $V$ and $W$ are vector spaces over a field $F$.

Let $W_1$ and $W_2$ be subspaces of a finite-dimensional vector space $V$.
Prove that $W_1$ and $W_2$ are independent if and only if $W_1\cap W_2=\{0\}$.

What is for you "independent linear subspaces"??

Tonio

Here is my work so far (well, it's how my Professor showed us during his office hours, so he probably made it incomplete).

$\forall w_1\in W_1$ and $\forall w_2\in W_2$ such that $w_1+w_2=0$, then $w_1=w_2=0$
If we suppose that $W_1,W_2$ are independent, this implies that $W_1\cap W_2=\{0\}$.
Suppose that $w\in W_1\cap W_2$,
Clearly, $w+(-w)=0$ (We can let $w_1=w$ and $w_2=-w$)
$\Rightarrow w=0$

Using the above, we now need to prove that $W_1\cap W_2=\{0\}\Rightarrow W_1,W_2$ are independent.
Suppose $w_1\in W_1$ and $w_2\in W_2$ and that $w_1+w_2=0$.
$\Rightarrow w_1=-w_2$ OR $w_2=-w_1$
$\Rightarrow w_1\in W_2$ OR $w_2\in W_1$

These imply that $w_1=w_2=0$, and that they are independent.

As such, $W_1$ and $W_2$ are independent.

-----

Tell me if my answer needs any clean-up or reordering.
.