# Basis of a Cartesian product

• Aug 25th 2010, 04:10 PM
mathematicalbagpiper
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....
• Aug 25th 2010, 11:21 PM
Swlabr
Quote:

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...
• Aug 26th 2010, 12:47 AM
Swlabr
Quote:

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!