Addition and Scalar multiplication in a vector space arc required to satisfy eig

Question

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.

Explanation / Answer

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.

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