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.

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.

we have the parent function given by,

1. the function is translated 5 units to the left.

so, we get,   becomes

2. the function is translated 5 units to the right.

so, we get,   becomes

3. the function is translated 5 units upwards.

so, we get,   becomes

4. the function is translated 5 units downwards.

so, we get,   becomes

5. the function is reflected across x-axis.

so, we get,   becomes

6. the function is reflected across y-axis.

so, we get,   becomes .

step-by-step explanation:

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