Vector space axiom

[mooshupork34]

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 \(\displaystyle x \oplus y =\) \(\displaystyle (x_1, x_2) \oplus (y_1, y_2) = (x_1 y_2, x_2 y_2)\), and as our scalar multiplication \(\displaystyle a \cdot x = a \cdot (x_1, x_2) = (a + x_1, a + x_2).\)


Verify the following vector space axiom:

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

 

[daon]

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 \(\displaystyle x \oplus y =\) \(\displaystyle (x_1, x_2) \oplus (y_1, y_2) = (x_1 y_2, x_2 y_2)\), and as our scalar multiplication \(\displaystyle a \cdot x = a \cdot (x_1, x_2) = (a + x_1, a + x_2).\)


Verify the following vector space axiom:

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

(x,y)+(a,b)=(xa,yb) =? (x,y).

What does (a,b) need to be? Your "zero" here may not be what you'd expect.