What is the difference between a Cartesian Product and a Direct Sum

[Hydro666]

Homework Statement

17. Let U = f(x; y; 0) : x 2 R; y 2 Rg, E1 = f(x; 0; 0) : x 2 Rg, and E3 = f(0; 0; x) :
x 2 Rg: Are the following assertions true or false? Explain.
(a) U + E1 is a subspace of R3:
(b) U  E1 is a direct sum decomposition of U + E1:
(c) U  E3 is a direct sum decomposition of R3:

Homework Equations

The Attempt at a Solution

I really need to know, from what my teacher told us about it, it seems like a Cartesian Product is adding dimensionality to two sets and that a Direct Sum is just adding every possible vector combination from two vector spaces together. But I really don't know

 

[annoymage]

Set A and B

Cartesian product
$$A\timesB=\left\{(a,b)|a \in A,\ b \in B\right\}$$

Direct Sum

$$A+B=\left\{a+b|a \in A,\ b \in B\right\}$$

something like that ;P

 

[Hydro666]

No, that's just an addition of subspaces, I'm talking about the "direct sum" its the little circle with the plus sign in it, I think its like a partisan cross product, but with vector spaces... I think