Linear Transformation from R^2 to R^3

[JG89]

Suppose a linear transformation $$T: R^2 \rightarrow R^3$$ was defined by $$T(a_1,a_2) = (2a_1, a_2 + a_1, 2a_2)$$. Now, for example, would I be allowed to evaluate $$T(3,8,0)$$ by rewriting (3,8,0) as (3,8)?

 

[Defennder]

Suppose a linear transformation $$T: R^2 \rightarrow R^3$$ was defined by $$T(a_1,a_2) = (2a_1, a_2 + a_1, 2a_2)$$. Now, for example, would I be allowed to evaluate $$T(3,8,0)$$ by rewriting (3,8,0) as (3,8)?

Not allowed. It's R^2 to begin with. In some cases, it might seem as though such practice were allowed, for example when you're working over the vector space of polynomial functions and have to add some polynomials which are not of the same degree, so the coefficients of the "missing" powers of x are treated as 0. However in such a case it's already implicitly understood that we usually omit writing 0x^3, 0x^4 for example even though they are there.

In your case, how ever, it is not clear cut as to why we should interpret (3,8,0) as (3,8). Why couldn't it be seen as (8,0) instead?

 

[JG89]

It seems to me that (3,8,0) and (3,8) represent the same location, if you interpret the coordinates geometrically. My high school math teacher said that this practice was allowed when evaluating cross products, so I thought it might have been okay here. For example, the cross product isn't defined in R^2. So if you wanted to find the cross product of (3,4) and (4,6), you would simply rewrite it as (3,4,0) and (4,6,0).

 

[Defennder]

Well it depends on the context. If you're an engineering student of course it makes sense to do so. But from a mathematical perspective it's not. It's only ok if it's understood to be intentionally omitted.

By the way you posted this in the Linear Algebra forums. Posting it elsewhere might net you a different answer. Don't try it though, since duplicate threads across different forums are frowned upon.

 

[JG89]

Thanks for the help (I was only concerned with the mathematical viewpoint).