To show that $T$ is linear if and only if $b = c = 0$, let's analyze the linearity properties of $T$. A transformation $T: V \to W$ is linear if for all $p, q \in V$ and scalars $\alpha, \beta$, we have:

$T(\alpha p + \beta q) = \alpha T(p) + \beta T(q).$

Here, $T$ is defined as: $T(p)(x) = 3p(4) + 5p'(6) + b p(1)p(2) + \int_{-1}^2 x^3 p(x) dx + c \sin(p(0)).$

Step 1: Check linearity components

We analyze each term of $T(p)$:

1. $3p(4)$: This term is linear because $p(4)$ depends linearly on $p$, and multiplication by a constant (3) preserves linearity.

2. $5p'(6)$: Similarly, $p'(6)$ is linear in $p$, as differentiation and evaluation are linear operations. Multiplication by 5 does not affect linearity.

3. $b p(1)p(2)$: This term is not linear in general because it involves the product $p(1)p(2)$. Specifically:

   • If $p$ and $q$ are polynomials, $p(1)p(2) + q(1)q(2) \neq (p + q)(1)(p + q)(2)$.
   • Thus, this term violates linearity unless $b = 0$.

4. $\int_{-1}^2 x^3 p(x) dx$: The integral of $x^3 p(x)$ over a fixed range is a linear operation. Hence, this term is linear.

5. $c \sin(p(0))$: The sine function is nonlinear unless $c = 0$, because $\sin(p(0))$ is not additive (i.e., $\sin(p(0) + q(0)) \neq \sin(p(0)) + \sin(q(0))$). For this term to preserve linearity, $c = 0$ is required.

Step 2: Combine results

The transformation $T(p)$ will be linear if and only if the terms $b p(1)p(2)$ and $c \sin(p(0))$ vanish, which happens when $b = 0$ and $c = 0$. Otherwise, these terms violate the additivity and homogeneity requirements for linearity.

Step 3: Proof

If $b = 0$ and $c = 0$:

The transformation becomes: $T(p)(x) = 3p(4) + 5p'(6) + \int_{-1}^2 x^3 p(x) dx,$ which is a sum of linear operations, and hence $T$ is linear.

If $b \neq 0$ or $c \neq 0$:

• If $b \neq 0$, the term $b p(1)p(2)$ introduces a nonlinear component.
• If $c \neq 0$, the term $c \sin(p(0))$ introduces another nonlinear component.

Thus, $T$ is not linear unless $b = c = 0$.

Conclusion

$T$ is linear if and only if $b = 0$ and $c = 0$.

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Let me know if you want a detailed example or more breakdown of any step. Here are 5 related questions for deeper understanding:

1. Can you provide an example where $T$ fails to be linear when $b \neq 0$?
2. How does the integral term ensure linearity despite involving $p(x)$?
3. Why is the sine function nonlinear in the context of $T(p)$?
4. Can you generalize the result to other transformations involving products of $p(x)$ values?
5. How would the proof change if $T$ were defined over a different domain or codomain?

Tip: Always isolate nonlinear components when testing a transformation for linearity.