Thread: Vector space axiom

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.

Originally Posted by faure72

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

are you sure it shouldn't be <x1, x2> + <y1, y2> = <x1*y1, x2*y2> ? otherwise, it doesn't work out nicely

Originally Posted by Jhevon

are you sure it shouldn't be <x1, x2> + <y1, y2> = <x1*y1, x2*y2> ? otherwise, it doesn't work out nicely

The <x1*y1, x2*y2> is what it's supposed to be. Sorry, forgot to include that multiplication symbol in there so I can see why it came off as unclear. The addition symbol is the vector addition symbol, though (the circle with the plus sign in it).

Originally Posted by faure72

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.


EDIT: The <x1*y1, x2*y2> is what it's supposed to be. Sorry, forgot to include that multiplication symbol in there so I can see why it came off as unclear. The addition symbol is the vector addition symbol, though (the circle with the plus sign in it).

ok, now what this question is asking for is the zero vector for this vector space. thus we want to find some element called 0 such that:

<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