Mark each of the following statements below as true or false

Mark each of the following statements below as true or false. Justify each answer. If E is the standard basis for R^n, then the E coordinate vector of an x in R^n is x itself. The correspondence [x]_E rightarrow x is called the coordinate mapping.

Solution

a. The statement is true. We know that the standard basis is a orthonormal basis in which each vector in the basis has only one nonzero entry, and that entry is equal to 1. The vectors are usually denoted e_i with i = 1,2,…,n, with n representing the dimension of the vector space that is spanned by this basis. Now, if E=       {e₁,e₂,…,e_n} is the standard basis foe Rⁿ and if A is the n x n matrix with e₁,e₂,…,e_n as its columns then the E coordinate vector of an x = ( x₁,x₂,…,x_n)in Rⁿ is Ax = x itself.

b. The statement is False. The correspondence x [x]_E is called the coordinate mapping. ëThe correspondence [x ]_E x is the inverse of the coordinate mapping.