Direct sum and direct product

The definition: We have two vector spaces [itex]V_1[/itex] and [itex]V_2[/itex] with bases [itex]\{e_i\}[/itex] and [itex]\{f_i\}[/itex]. A basis for the direct product space [itex]V_1\otimes V_2[/itex] can be taken as [itex]\{e_i\otimes f_j\}[/itex]. So an element w of this space would look like:



[tex]w=A^{ij}e_i\otimes f_j [/tex]



Now if we take [itex]V_1[/itex] to be the x-axis, and [itex]V_2[/itex] to be the y-axis, we can say that the tensor product space is the y=x line. Since [itex]w=A\;\;\hat{x}\otimes \hat{y} [/itex]





For the direct sum space [itex]V_1\oplus V_2[/itex], we take as basis: [itex]\{e_1,e_2,....;f_1,f_2,...\}[/itex]. And so the space as in the previous example consists of the entire [itex]R^2[/itex].



Are the last two statements correct? If not, what is my mistake?



Thanks!