Addition and Scalar multiplication in a vector space arc req

Addition and Scalar multiplication in a vector space arc required to satisfy eight rules. List them. Prove that the space is a subspace or demonstrate which of the requirements for a subspace are violated: The set of vectors in R^2 whose components are positive or zero. The set of functions continuous on the open interval from zero to one. Given the system: x + 2y - 2z = b_1 2x + 5y - 4z = b_2 4x + 9y - 8z = b_3 Find the solvability condition. Find the basis for the column space. Find a particular solution for when b = [-1 -2 0] Find the basis for the null space. Write the complete solution to the system. Find the basis for the row space. Find the basis for the left null space.

Solution

1. Let V be a non-empty set and + be a binary composition on V. Let (F, +, .) be a field and let * be an external composition of F with V.

+ is known as vector addition and * is known as scalar multiplication.

V is said to be a vector space over the field F if the following conditions are satisfied.