mooshupork34

05-04-2007, 08:18 PM

If anyone could explain how the following linear algebra problem is done, it would be greatly appreciated as I am studying for a test!

Here is a vector space candidate. We have as our set R^2, as our vector addition x \oplus y = (x_1, x_2) \oplus (y_1, y_2) = (x_1 y_2, x_2 y_2), and as our scalar multiplication a \cdot x = a \cdot (x_1, x_2) = (a + x_1, a + x_2).

Verify the following vector space axiom:

There exists an element 0 such that for any x in the proposed vector space, x \oplus 0 = x = 0 \oplus x.

Here is a vector space candidate. We have as our set R^2, as our vector addition x \oplus y = (x_1, x_2) \oplus (y_1, y_2) = (x_1 y_2, x_2 y_2), and as our scalar multiplication a \cdot x = a \cdot (x_1, x_2) = (a + x_1, a + x_2).

Verify the following vector space axiom:

There exists an element 0 such that for any x in the proposed vector space, x \oplus 0 = x = 0 \oplus x.