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Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They play a significant role in algebra and calculus, particularly when exploring linear transformations. Here, we focus on polynomials of degree two, represented in the form:

• Second-degree polynomial: \( a_0 + a_1 x + a_2 x^2 \)

Understanding how these polynomial expressions transform is crucial in determining if a function, like \( T \) given in the exercise, is a linear transformation.
Linear transformations are operations that map one vector space to another, in a way that preserves the operations of addition and scalar multiplication intrinsic to those spaces.
The function \( T \) must, therefore, confirm to these properties while transforming polynomial functions from the polynomial space \( P_{2} \), which consists of all polynomials of degree two or less, back into itself.

The addition property is a fundamental aspect of linear transformations. This property asserts that for a function \( T \), if it is indeed a linear transformation, the result of adding two polynomials and then applying \( T \) should be the same as applying \( T \) to each polynomial individually and then adding the results.
Consider two generic polynomials of degree two, \( u = a + b x + c x^2 \) and \( v = d + e x + f x^2 \). Their sum is another polynomial:

• \( u+v = (a+d) + (b+e)x + (c+f)x^2 \)

Now, applying \( T \) to this sum should equal the sum of \( T(u) \) and \( T(v) \).
In the solution steps, it was shown that:

• \( T(u+v) = (b+e) + 2(c+f)x \)
• \( T(u) + T(v) = (b+e) + 2(c+f)x \)

This ensures the addition property holds, confirming this part of \( T \)’s linear transformation criteria.

The scalar multiplication property is another essential condition for a function to be recognized as a linear transformation. This property stipulates that scaling a polynomial by a constant (or scalar) and then applying the function \( T \) should yield the same result as first applying \( T \) to the polynomial and then scaling the result.For a polynomial \( u = a + bx + cx^2 \) and a scalar \( k \), scaling \( u \) gives:

• \( ku = ka + kbx + kcx^2 \)

Apply \( T \) to get:

• \( T(ku) = kb + 2kc x \)

If we first apply \( T \) to \( u \), and then scale:

• \( T(u) = b + 2c x \)
• \( kT(u) = kb + 2kc x \)

The results, \( T(ku) \) and \( kT(u) \), match, verifying the scalar multiplication property for \( T \). This confirms that \( T \) upholds both linear transformation requirements.