# map

1. ### Prove that the map is closed

I need to prove that map: $$C^n\ni (z_1,...,z_n)\mapsto z_1\in C$$ is closed, and I wonder how to take a closed set in $(z_1,...,z_n)$. Can you give me a hint? Thank you.
2. ### Question regarding the Image of a linear map

Let $V$ be a vector space of finite dimension, and let $T: V\rightarrow V$ a linear map. Why is it true that there must me a $k$ for which $ImT^k=ImT^{k+1}$? I can see why it's true when $T$ is not invertible, since in this case, $T$ is nilpotent of some order, and this $k$ is that order. But...
3. ### Help with proving that kernel is a proper subset of the image of a linear map

Let $V$ be a vector space of dimension 3, and let $T:V\rightarrow V$ be a nilpotent linear operator of order 3. I need to prove the following: 1. Suppose $v\in V$ is a vector for which $T^2(v)\neq 0$, prove that the set $B=\{v,T(v),T^2(v)\}$ is a basis for $V$. 2. Prove that \$Ker(T)\subset...
4. ### Calculating the Surface Area for a Fictional Map

Okay, so I would like to figure out the surface area of a fictional continent i'm making for a fictional fantasy map. Ive mad an example continent real quick that is simple. The question I have is if 42 pixels wide is equal to 103 kilometers, then how many kilometers is 1 pixel equal to? and...
5. ### When it's a "function" and when it's a "map"

Greetings, This is another potentially stupid question, but... Are the terms "function" and "map" fully interchangeable. If not in which are the areas of mathematics that benefit from this distinction. I'm asking because for the first time I've read some mathematical texts on my native...
6. ### Riemannian geometry exponential map and distance

Hi all. For some reason I have been having a lot of difficulty with this problem in Peter Petersen's text. The problem is Prove: d(exp_p(tv), exp_p(tw)) = |t||v-w| + O(t2 ) The exponential map is the usual geodesic exponential map. And d(p,q) is the infinum of the lengths of all curves starting...
7. ### Find minimum AND-OR expression using a Karnaugh Map?

Use a Karnaugh Map to find the minimum AND-OR expression for x(a,b,c): For part a the answer is: x(a, b, c) = ac + b'c' How do I get that answer? I know from the problem that x=1 at rows 0, 4, 5, and 7 of the respective truth table (which is not given), however I thought x can only equal 1...
8. ### Karnaugh Map Problem (Logic)

I don't know if this would be the right forum to post this at, but it seems this place is full of all kinds of smart. If anyone's familiar with Project Lead the Way it might help more also. There's a problem in it (2.2.5) that uses four sensors on a fire place and three out of four must have a...
9. ### Open map theorem

Hello! I wanted to ask something about functional analysis but i looks that this topic doesn't exist. I hope it is ok to publish it here. Well, I wanted to proove that: Let X and Y be two Banach spaces and "f" a map between the, such tahta it is linear, continuous and surjerctive. Then: If K is...
10. ### kenrnal of a map for group action

Okay, given a function for a group action of the group G acting on a set A (with a fixed point a of A): \phi _a : G \times \{ a \} \to A. One definition I have for the kernal is ker \phi _a = \{ (g, a) | \phi _a (g, a) = 1_A ~ \forall g \in G \}. But the definition in the group action section...
11. ### map (m,n) to N

This is a reply to Post QXQ countable sets which the system won't let me rreply to. let g(m,n) = (2m-1)2^(n-1) Is this a 1-1 mapping to N? If it is, it proves the rational numbers are countable since (m,n) can be interpreted as rational numbers. g maps (m,n) to a unique even positive...
12. ### Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Properties of the map I

I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, the set \mathcal{I} (A) is defined in the following text on page 660: (see attachment)...
13. ### Algorithm to traverse a map (comp sci)

Given a topographical map that lists the altitude along the direct road between any two neighboring cities, and two cities a and b, how would you develop a linear time algorithm that finds a route from a to b that minimizes the maximum altitude? Roads can be traveled in both directions. I have...
14. ### Homomorphism of rings, inverses map to inverse?

Hi, If \phi: R \to S is a homomorphism between two rings R and S. Is it true that: If a \in R is a unit then \phi(a^{-1})=\phi(a)^{-1}, or more generally do units have to map to units? Thanks
15. ### Scale Ratio

A map has a scale of 1:1000. (i) What distance on the ground (in meters) is represented on the map as (a) 2 cm (b) 7.6 cm (ii) What distance on the map represents (a) 100 m (b) 2.6 km ------------------ Can anyone help me to find the answer step by step...
16. ### Gauss map and its derivative

Compute the Gauss map and its derivative for the cone parameterized as follows X(u,v) =(vcosu, vsinu, v) What does the image of the Gauss map on the sphere look like?
17. ### inverse stereographic projection map

Consider the sphere S= {(x,y,z)} ∈ R^3 | x² + y²+ z²=1 }. Let N=(0,0,1) be the north pole of the sphere. The inverse stereographic projection map s is a homomorphism s:R^2 s →S-N defined by mapping the point (x,y) ∈ R^2 to the point on S that lies on the line connection (x,y,0) to N in R^3...
18. ### Linear Algebra Subspace regarding linear map

Hi, i'm having trouble starting this proof and how to prove the ideas necessary. Let V and W be finite-dimensional vector spaces over F. Given T is in L(V,W), show that there is a subspace U of V such that the following are true: U(intersection)null(T)= {0} and range(T)={Tu:u in U}. Thank you
19. ### Linear map help

i need help with part a and b, not sure if they are or are not linear or not, for part c) can i set x = 0 and y = , which gives me 2, which does not equal 0, so part c is not a linear map? also part d, is not a linear map because if a =0 , b = 0, and c= 0, than we get 0, in the...
20. ### Rotation matrices and exponential map

Hi guys, I'm having some trouble with rotation matrices. Basically, I want to show that any matrix R(n \alpha) \in SO(3), specified by an angle \alpha and a unit vector n\in S^2 can be written as R(n \alpha) = exp (\alpha \sum n_i J_i ), where J_i is a 'basis' matrix for \mathfrak{so}(3)...