Prove Corollary 6.2. If L : V ? W is a linear transformation of a vector space V into a vector space W and dim V=dim W, then the following statements are true: (a) If L is one-to-one, then it is onto. (b) If L is onto, then it is one-to-one.
Accepted Solution
A:
Answer with explanation:Given L: V\rightarrow W is a linear transformation of a vector space V into a vector space W.Let Dim V= DimW=na.If L is one-one Then nullity=0 .It means dimension of null space is zero.By rank- nullity theorem we have Rank+nullity= Dim V=nRank+0=nRank=nHence, the linear transformation is onto. Because dimension of range is equal to dimension of codomain.b.If linear transformation is onto.It means dimension of range space is equal to dimension of codomainRank=nBy rank nullity theorem we have Rank + nullity=dimVn+nullity=nNullity=n-n=0Dimension of null space is zero.Hence, the linear transformation is one-one.