Linear Algebra - one to one and onto question

[zeion]

Homework Statement

Determine whether each of the following mappings f is onto or one-to-one. Is f an isomorphism?

1) f maps R2 to R3 defined by f(x, y) = (x, y, x+y)
2) f maps R3 to R(1x3) defined by f(x,y,z) = [x^2, y^2, z^2]
3) f maps R4 to P2(R) defined by f(a,b,c,d) = a+(b-c)x+dx^2

Homework Equations

The Attempt at a Solution

I'm not sure how to do these.. I understand somewhat about one-to-one and onto, but these notations kind of confuse me.. For onto, do I just need to look at the outcome and see if it spans the space? And for one-to-one, I look at rather every component from the source appears in every component in the result?

1) Not one-to-one, onto.
2) Not one to one, onto.
3) Not one to one, onto.

 

[Mark44]

Homework Statement

Determine whether each of the following mappings f is onto or one-to-one. Is f an isomorphism?

1) f maps R2 to R3 defined by f(x, y) = (x, y, x+y)
2) f maps R3 to R(1x3) defined by f(x,y,z) = [x^2, y^2, z^2]
3) f maps R4 to P2(R) defined by f(a,b,c,d) = a+(b-c)x+dx^2

Homework Equations

The Attempt at a Solution

I'm not sure how to do these.. I understand somewhat about one-to-one and onto, but these notations kind of confuse me..

You need to understand the definitions of these terms more than just somewhat. Go back and look at the definitions of one-to-one and onto, and any examples there are in your book.

For onto, do I just need to look at the outcome and see if it spans the space?

Pretty much. More precisely, if f is a map from U to V, f is onto V if, for any v in V, there is a u in U such that f(u) = v. IOW, no matter what thing you pick in the output space, there is a thing in the input space that maps to it. For example, if f:R --> R is defined by f(x) = x², f is not onto, since -1 is in R (the output space), but there is no real number x in R (the input space) such that f(x = -1.

And for one-to-one, I look at rather every component from the source appears in every component in the result?

That's not how one-to-one-ness is defined. One definition is that if a != b, then f(a) != f(b). Using the same function as my previous example, f is not one-to-one, since f(2) = f(-2).

1) Not one-to-one, onto.
2) Not one to one, onto.
3) Not one to one, onto.