# Thread: Basis of a Cartesian product

1. ## Basis of a Cartesian product

Let $\displaystyle U, V$ be finite dimensional linear spaces and let $\displaystyle \{u_1,...,u_n\}$ be a basis for $\displaystyle U$ and let $\displaystyle \{v_1,...,v_m\}$ be a basis for $\displaystyle V$. Let $\displaystyle W=U\times V$. Construct a basis for $\displaystyle W$.

I'm probably making this way harder than it really is. The only idea I came up with was

$\displaystyle \{(u_i, v_j) | 1\leq i\leq n, 1\leq j\leq m\}$

But I know that's wrong....

2. Originally Posted by mathematicalbagpiper
Let $\displaystyle U, V$ be finite dimensional linear spaces and let $\displaystyle \{u_1,...,u_n\}$ be a basis for $\displaystyle U$ and let $\displaystyle \{v_1,...,v_m\}$ be a basis for $\displaystyle V$. Let $\displaystyle W=U\times V$. Construct a basis for $\displaystyle W$.

I'm probably making this way harder than it really is. The only idea I came up with was

$\displaystyle \{(u_i, v_j) | 1\leq i\leq n, 1\leq j\leq m\}$

But I know that's wrong....
Think of an example. Take your favourite 2-dimensional vector space and cross it with your favourite 3-dimensional vector space. Can you think of a basis for this new space?

Also, I would recommend contemplating how many elements your new basis will have. Hopefully the example will make this clear, but if not...think about it...

3. Originally Posted by Isomorphism
To solve your problem, consider a set with the elements of the form -Blank-.
Don't just give the answer away! Rule 14!