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The problem statement has been attached.

To show that T : V →R is a linear function

It must satisfy 2 conditions:

1) T(cv) = cT(v) where c is a constant

and

2) T(u+v) = T(u)+T(v)

For condition 1)

T(cv)=∫cvdx from 0 to 1 (I don't know how to put limits into the integral.

cT(v)=c∫vdx (pulling a constant out of an integral)

cT(v)=cT(v)

I think I did the work for condition 1 right?

For condition 2)

T(u+v) where u=u1 and v=v1

T(u+v)=∫(u+v)dx

T(u+v)=∫(u)dx + ∫(v)dx (this is called the sum rule of integration, I think).

T(u+v)=T(u)+T(v)

Did i do this right?

To show that T : V →R is a linear function

It must satisfy 2 conditions:

1) T(cv) = cT(v) where c is a constant

and

2) T(u+v) = T(u)+T(v)

For condition 1)

T(cv)=∫cvdx from 0 to 1 (I don't know how to put limits into the integral.

cT(v)=c∫vdx (pulling a constant out of an integral)

cT(v)=cT(v)

I think I did the work for condition 1 right?

For condition 2)

T(u+v) where u=u1 and v=v1

T(u+v)=∫(u+v)dx

T(u+v)=∫(u)dx + ∫(v)dx (this is called the sum rule of integration, I think).

T(u+v)=T(u)+T(v)

Did i do this right?