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laminatedevildoll

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I've uploaded a document which I am currently working on. I would like to verify if I am doing these problems correctly. Thank you.

In the first attachment (3.4b)

For 1.

a. 4x^3-2x

b. T(P)=0

ker T={C:C [tex]\in[/tex]R}

Im T = {P|P is less than degree 3 or less}

c. T is not one to one because P is a constant. T is not onto because it's degree less than 3. I am not sure if I am proving this right. I'd appreciate some help.

For 2.

a. 1/5x^5-1/3x^3+C

b.

ker T={C:C [tex]\in[/tex]R}

Im T = {P|P is less than degree 5 or less}

T is not one to one because C=0. But, T is on-to. How do I prove this?

In the second attachment (3.4a) part 4.

I proved that T is one to one because T(f)=T(g), f=g How do I prove that this is on-to?

In the first attachment (3.4b)

For 1.

a. 4x^3-2x

b. T(P)=0

ker T={C:C [tex]\in[/tex]R}

Im T = {P|P is less than degree 3 or less}

c. T is not one to one because P is a constant. T is not onto because it's degree less than 3. I am not sure if I am proving this right. I'd appreciate some help.

For 2.

a. 1/5x^5-1/3x^3+C

b.

ker T={C:C [tex]\in[/tex]R}

Im T = {P|P is less than degree 5 or less}

T is not one to one because C=0. But, T is on-to. How do I prove this?

In the second attachment (3.4a) part 4.

I proved that T is one to one because T(f)=T(g), f=g How do I prove that this is on-to?