Calculus Analysis for Newbies: Point vs Vector in Rn

[flyingfrog]

hi, I am a newbie in calculus analysis, I have a question when I reading the definition of Derivatives in Rn: I don't know what is the different of point and vector in Rn ? and why some time the definition use point (c) and sometime use vector (u)?

 

[HallsofIvy]

We can do "arithmetic" with vectors (add vectors, multiply vectors by scalars) but not with points. Of course, once we have chosen a coordinate system, we can associate a vector with each point (the vector from the origin to that vector) but we can work with points without a coordinate system.

 

[flyingfrog]

Thank you very much, and I have another question about the proof of derivative in Rn, it said define y0 = f(b) - f(a) , for y0<>0, define y1 = y0/||y0|| then it define H(x) = f(x) * y1, y1 called the unit vector, but then my confusion is how come H(b) - H(a) = {f(b) - f(a)} * y1 = ||f(b) -f(a)||?

and mean while, I found during the analysis class, I am lacking of linear alegbra knowledge, do you have any good book to recommend? again, thank you very much for the help

 

[Rasalhague]

Does this make it clear?

$$H(b)-H(a)=(f(b)-f(a)) \cdot y_1 = \frac{y_0 \cdot y_0}{\left \| y_0 \right \|}$$

$$= \frac{\left \| y_0 \right \|^2}{\left \| y_0 \right \|} = \left \| y_0 \right \| = \left \| f(b)-f(a) \right \|$$

 

[Rasalhague]

hi, I am a newbie in calculus analysis, I have a question when I reading the definition of Derivatives in Rn: I don't know what is the different of point and vector in Rn ? and why some time the definition use point (c) and sometime use vector (u)?

In my experience, the words "point" and "vector" are used interchangeably in this context. To say that (4,-1,19) is a point in R³ is to say exactly the same as that (4,-1,19) is a vector in R³.

Rⁿ, with the standard rules for addition and scalar multiplication, is a vector space; so points in this space can be called vectors.

So if an author wants to make a distinction of their own between the two terms, they would have to explain what they meant.

The word "point" is also used, more generally, for elements of other kinds of space, spaces which are not necessarily vectors spaces, such as topological spaces, metric spaces, affine spaces, manifolds. The word "vector" is reserved for points of a vector space, that is, a mathematical structure which obeys the vector space axioms. (Don't worry if you haven't studied those other kinds of space yet; they're just different mathematical structures, each kind defined by its own axioms.) Where a particular vector space is understood from the context, "vector" may mean a vector of that particular vector space, such as a vector in Rⁿ.