nullhomotopic

I am having problems proving this fact. A space X is contractible if and only if every map f:X to Y (Y is arbitrary) is nullhomotopic. Similarly show X is contractible iff ever map f:Y to X is nullhomotopic. In the first case if Y=X then see that the identity map on X is nullhomotopic. But Im not sure how to proceed for the rest.

a space whose fundamental group is non-abelian

Give an example of a space whose fundamental group is non-abelian

Module Theory: Vector space over complex numbers

Problem 1. Let V be a finite-dimensional complex vector space. Then V is also a vector space over real numbers R. Show that dimV ( over R) = 2*dimV(over complex C). Hint: If B={v1, v2, ..., vn} is a basis of V over C, show that B'={v1, ..., vn, iv1, ... ivn} is a basis of V over R. Problem 2. ( extend problem1) Let L be a field and let K be a subfield of L. If V is a vector space over L, then it is also a vector space over K. Prove that dim_K(V)=[L : K]dim_L(V) where [L:K] = dim_K(L) is the dimension of L as a vector space over K. (Note that it is not assumed that dim_K(L)

Algebraic Theory

1. Do the operations and simplify. (a^-2b^3)/(x^-8y) . (a^6b^2)/(x^8y^4) 2. Do the operations and simplify. (x^2 + 8x + 16)/(x) . (x^2 - 4x)/(x^2 - 16) 3. Do the operations and simplify. (a + 5)/(25 - a^2) . (5a - 25)/(5a + 25) 4. Simplify the expression. (x2 - 36)/(x^2 - 64) ÷ (x + 6)/(x - 8) 5. Simplify the cube root. ^3√[8a^30b^21] 6. Simplify and combine like radicals . All variables represent positive numbers. 8^5√[3y] - 2^5√[3y] 7. Simplify and combine like radicals. √[45] - √[245] 8. Multiply and simplify the result. √[125b] and √[5b^3n] 9. Multiply and simplify the... click for more

Text Book : Advance Calculus, Author: - Taylor and Menon

In page number 160 i need following questions following questions to be answered: 2, 4, 7 & 11. Please mention each and every step.