Question

Which transformations if any are one to one or onto? 2) T is the orthogonal projection of R^2 onto the y axis or T is the orthogonal projection of R^3 on to the xz plane?

Answer 1

How would you go about this question? I don't really understand the concept of onto and one to one

Answer 2

one to one
\[T(v)=T(u)\Rightarrow u=v\]

Answer 3

onto T from V to W is onto means if w is in W then there is a v in V with T(v)= w

Answer 4

so for example if y is on y axis then (x,y) in R^2 is projected on to y. so it is certainly onto
but not one to one, since for example both (2,3) and (1,3) get sent to 3

Answer 5

The projections would only be onto if the target space is smaller than the source space. The transformation from R^3 to the xz plane is describing a transformation from R^3 to a plane within R^3. It is not a transformation from R^3 to R^2.

Answer 6

It does not even make sense to consider a plane within R^2. It only makes sense in R^3 in this case.

Answer 7

The same holds for the projection to the line. I think I would say that neither transformation is one to one or onto.