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.
<x1, x2> + 0 = 0 + <x1, x2> = <x1, x2>
let the zero vector be <a,b>
we want <a,b> + <x1,x2> = <x1, x2>
now, <a,b> + <x1, x2> = <a*x1, b*x2>
for <a*x1, b*x2> = <x1, x2> we must have,
a*x1 = x1
=> a = 1
and b*x2 = x2
=> b = 1
so our zero vector is <1, 1>
check the other way,
<x1, x2> + <1, 1> = <x1*1, x2*1> = <x1,x2>
thus our element zero is <1,1> for this vector space