Question

Lemma. If two vector spaces have the same dimension then they are isomorphic Proof. To show that any two spaces of dimension n are isomorphic, we can simply show that any one is isomorphic to R. Then we will have shown that they are isomorphic to each other, by the transitivity of isomorphism (which was established in the first Theorem of this section) Theorem 1 Isomorphism is an equivalence relation among ctor spaces Let v be n--dimensional. Fix a basis B-(81, write , β.) for the domain V. Given v E V, we may and the representation of with respect toB V1 may be considered a function f.V → R". This is indeed a function (i. e., is well defined) because given any vector v, there is exactly one way to write ץ as a linear combination of vectors in B, and Rep V is completely determined by this representation. We will show that f is an isomorphism. This entails showing that f is one--to--one, onto, and preserves structure WI First we show that f is onto. Choose a vector- D E R". Then v-w1βι + + wnA, E V, and f(7): w. So f is onto To show f is one-to-one, suppose 급,すe V, and that f(띠 = f(7). (See question below.) Writing u-ul βι + + unf, and v-v1 βι + + vf , we have u1 ) Repu- Normal XML tags Source Un V1 f u) fv) implies ul-V1 Un Vn and this impliesv. Proving that f preserves structure is facilitated by the last Lemma of the previous section. Let u-u1A + c1, c2 e R, we must show that f(qu + cy-cif(动+ c2f(7). A routine calculation shows that both of these are + u' A, and v-v, βι + … + v, β be vectors in v and let Lemma 1 For any map f:V ollowing are equivalent. W between vector spaces the .f preserves structure;that is fv+)-v)+ 2. f preserves linear combinations of two vectors: 3. f preserves linear combinations of any finite number %) and fev)-cf(V) Ciun C2Vn of vectors: Question. To prove that f is one-to-one, what must be concluded after we suppose that f(î -(v)?

Answer 1

To pro ve f is one - one we hau e to ie· we can say logically and thus KeYH)=o clom ain Co-domain Null 3pqce ange spa ce.