# Function domain and range

1. Jul 14, 2009

### angryfaceofdr

Say we have a function, $f(x)=x^3$

one would say "$f$ is a function from $\mathbb{R}$ to $\mathbb{R}$" or $f: \mathbb{R}\to\mathbb{R}$

Then say we have a vector function, $\vec{g}(t)=<t^2+1,t>$.

How would one use the above notation? Would it be $\vec{g}: \mathbb{R}\to\mathbb{R}^2$?

And could one say that $\mathbb{R}^2$ is the same as the vector space $\mathbb{R}^2$?

What is the difference between a set of vectors and a set of points?

2. Jul 14, 2009

### trambolin

3. Jul 15, 2009

### angryfaceofdr

So . . .

Is this right then?

4. Jul 15, 2009

### angryfaceofdr

(from page 2)
So if a vector is an ordered pair of points, does this mean that a collection of vectors is a collection of points?

5. Jul 15, 2009

### HallsofIvy

Staff Emeritus
No, that means a collection of vectors is a collection of ordered pairs of points!

If you mean to think of the "pairs of points" as simply a collection of points, you lose the "ordering" which is an important part of vectors. For example, if P, Q, and R are points and your collection of vectors is {(P,Q), (P,R)}, that is very different from the collection of points {P, Q, R}. Note, for example that the collection of vectors {(Q,P)(Q,R)} contains different vectors than the first example but would "reduce" to the same collection of points, {P, Q, R}.

6. Jul 15, 2009

### angryfaceofdr

Is my post from #3 correct????

Ok, so in linear algebra I recall reading something along the lines of
Theorem 1:
Suppose , $\vec{u}$ and $\vec{v}$ are vectors in $\mathbb{R}^n$. We say that $\vec{u}$ and $\vec{v}$ are orthogonal if $\vec{v}\cdot \vec{u}=0$.

So... say we are "in" $\mathbb{R}^2$. Is is true that $\mathbb{R}^2 =\mathbb{R}$ x $\mathbb{R}=\{ (x,y)|x\in \mathbb{R}$ and $y \in \mathbb{R} \}$?

Then say we have a point (2,3), is it true that (2,3) $\in \mathbb{R}^2$?

Then is it also true that the vector $\vec{u}=<2,3> \in \mathbb{R}^2$?

7. Jul 15, 2009

### angryfaceofdr

http://en.wikipedia.org/wiki/Euclidean_space

wiki says :

"For any non-negative integer n, the space of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted R^n and sometimes called real coordinate space. An element of R^n is written $\vec{x}=(x_1,x_2, \ldots x_n)$ . . ."

Does this mean that since (2,3) is a point and does not have the "ordering" that vectors have, the point (2,3) is NOT an element of R^2, but the vector $\vec{u}=<2,3>$ IS an element of R^2? (According to wiki an element of R^n needs to be a vector)

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ISSUE #1 IN MY HEAD: if $\vec{g}: \mathbb{R}\to\mathbb{R}^2$ (see post #1)
and (x,y) is some arbitrary point, (and thus from my understanding as of now this point (x,y) is not an element of R^2)

then how would one write the function definition of $h(x,y)=x^2+y^2$?
I couldn't write $h:\mathbb{R}^2 \to \mathbb{R}$ since (x,y) isn't a vector? Or do you make it a vector, then plug it into $h$?

Last edited: Jul 15, 2009