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The concept of additivity is essential in determining whether a transformation is linear. Additivity implies that when you apply a transformation to the sum of two functions, it should be the same as applying the transformation to each function individually and then adding their results. In mathematical terms, if you have two functions, say, \(f\) and \(g\), from a space \(\mathscr{F}\), a transformation \(T\) exhibits additivity if:

• \(T(f + g) = T(f) + T(g)\)

This relationship highlights that the transformation does not distort the addition. Instead, it respects the structure of the space it operates in.
In the given exercise, the transformation \(T(f) = f(c)\) is checked for additivity. When considering two functions \(f\) and \(g\), their sum evaluated at \(c\) is \((f+g)(c) = f(c) + g(c)\). This confirms the additivity property for our transformation, showing that \(T(f+g)\) indeed equals \(T(f) + T(g)\).
Understanding additivity ensures that a function behaves predictably when handling multiple inputs, making it a building block for linearity.

Scalar multiplication is another key property of linear transformations. It involves examining how a function behaves when its input is multiplied by a scalar value. For a transformation \(T\) to be linear with respect to scalar multiplication, it must satisfy the condition:

• \(T(kf) = kT(f)\)

Here, \(k\) is a scalar, and \(f\) is a function from the space \(\mathscr{F}\).
Imagine multiplying a function \(f\) by a scalar \(k\) before applying the transformation. The result should be the same as applying the transformation to \(f\) first and then multiplying the outcome by the scalar \(k\).
In our exercise, when \(T(f) = f(c)\), and \(k\) is applied to \(f\) giving \(kf\), it becomes \(T(kf) = (kf)(c) = kf(c)\). This means that \(T(kf)\) equals \(kT(f)\), confirming that scalar multiplication is preserved. This property allows the function to reflect scaled changes without altering the inherent operations within the space, offering a predictable manner in how transformations react to scalars.

When examining linear transformations, function evaluation plays a pivotal role. Function evaluation refers to how functions are selected and processed by a transformation. Essentially, it is the action of applying a transformation's rule to functions from a particular space.
In our given transformation \(T: \mathscr{F} \rightarrow \mathbb{R}\) defined as \(T(f) = f(c)\), the rule is straightforward—take a function \(f\) from the space \(\mathscr{F}\) and evaluate it at a chosen fixed point \(c\). For any specific function, \(T(f)\) gives a real number, thereby shrinking the functional output into the real number line.
The beauty of function evaluation in a linear transformation context is its consistency and predictability. It ensures that regardless of the specific function selected, applying the rule will yield consistent outputs based on the input and fixed components involved. This critical component checks whether the transformation holds linear properties, such as additivity and scalar multiplication, by constraining evaluation to a consistent method.