Question

Proofs are not necessary

Exercise 6.8.12. Determine if the following statements are true or false. If a statement is true, prove it. If a statement is false, give a counterexample or some other proof showing it is false. Unless otherwise specified, let V and W be a finite-dimensional vector space over field F, let (v1, ..., Un} be a basis of V, let {1,...,n} be a subset of W (possibly with repeated vectors), and let 6: V W be the linear transformation defined as o(aut...+40) = 21w1+...+a,w, for all a, ..., EF. A) If w1 = ... =wn = 0, then dim(im()) = 0. B) If wu = ... =wn = w and w 70, then dimim(6)) = 1. C) {W1, ..., Wn} spans im(o). D) dim(im(o)) = n. E) If is injective and surjective, then {1,...,n} is a basis of W. F) If {1,..., wn} is a basis of W, then o is injective and surjective. G) If {w1,..., Wr} spans W, then ø is injective. H) If {W1, ..., wn} is linearly independent, then o is surjective.

Answer 1

A] B c) D) True True True Falle E e G) H True True True True