Vector space axiom

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mooshupork34

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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⊕y=\displaystyle x \oplus y = (x1,x2)⊕(y1,y2)=(x1y2,x2y2)\displaystyle (x_1, x_2) \oplus (y_1, y_2) = (x_1 y_2, x_2 y_2), and as our scalar multiplication a⋅x=a⋅(x1,x2)=(a+x1,a+x2).\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, x⊕0=x=0⊕x\displaystyle x \oplus 0 = x = 0 \oplus x.
 
mooshupork34 said:
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⊕y=\displaystyle x \oplus y = (x1,x2)⊕(y1,y2)=(x1y2,x2y2)\displaystyle (x_1, x_2) \oplus (y_1, y_2) = (x_1 y_2, x_2 y_2), and as our scalar multiplication a⋅x=a⋅(x1,x2)=(a+x1,a+x2).\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, x⊕0=x=0⊕x\displaystyle x \oplus 0 = x = 0 \oplus x.
Click to expand...

(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.
 
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