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.

Question

Layne Potts

Mathematics

12 June, 09:23

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.

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Damon Hickman · Answer 1

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=n

a. 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=n

Rank+0=n

Rank=n

Hence, 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 codomain

Rank=n

By rank nullity theorem we have

Rank + nullity=dimV

n+nullity=n

Nullity=n-n=0

Dimension of null space is zero. Hence, the linear transformation is one-one.