Thread: Linear algebra question

If anyone could explain how the following is done, it would be greatly appreciated!

*Note that (+) denotes the vector addition symbol (which is displayed in textbooks as a circle with a plus sign inside it)

Here is a vector space candidate. We have as our set, R^2, as our vector addition x (+) y = (x1, x2) (+) (y1, y2) = (x1y1, x2y2) and as our scalar multiplication a * x = a * (x1, x2) = (a + x1, a + x2). Verify the following vector space axiom.

a * (x (+) y) = (a * x) (+) (a * y)

Originally Posted by faure72

If anyone could explain how the following is done, it would be greatly appreciated!

*Note that (+) denotes the vector addition symbol (which is displayed in textbooks as a circle with a plus sign inside it)

Here is a vector space candidate. We have as our set, R^2, as our vector addition x (+) y = (x1, x2) (+) (y1, y2) = (x1y1, x2y2) and as our scalar multiplication a * x = a * (x1, x2) = (a + x1, a + x2). Verify the following vector space axiom.

a * (x (+) y) = (a * x) (+) (a * y)

Let x be the point (b, c) and y the point (d, e).

Then
x (+) y = (bd, ce)
a * (x (+) y) = (a + bd, a + ce)

Now
a * x = (a + b, a + c)
a * y = (a + d, a + e)

(a * x) (+) (a * y) = ((a + b)(a + d), (a + c)(a + e))

= (a^2 + (b + d)a + bd, a^2 + (c + e)a + cd)

I can't see how the two can be equal if x and y belong to R^2 without some other restriction.

-Dan