Question

Which of the following transformations are linear? A. T(f(t))=∫ −9/8 f(t)dt from P 4​to R B. T(f(t))=f ′ (t)+8f(t)+4 from C[infinity] to C[infinity] C. T(f(t))=f(2) from P​ to R D. T(f(t))=3(f(t)) 2 +7f(t) from C[infinity] to C[infinity] E. T(f(t))=f ′ (t)+3f(t) from C[infinity] to C[infinity]

Answer 1

Among the options given, the linear transformations are B and E, which involve differentiation and algebraic operations that preserve additivity and homogeneity, key properties of linear transformations.

Step-by-step explanation:

The student asked which of the given transformations are linear transformations. A transformation T is considered linear if it satisfies two properties for all vectors f(t) and g(t) in its domain and any scalar α:

• T(f(t) + g(t)) = T(f(t)) + T(g(t)) (additivity)
• T(αf(t)) = αT(f(t)) (homogeneity)

Let's evaluate the options given:

• A is not a linear transformation because the integration limits are fixed and thus it does not satisfy the additivity and homogeneity property for any pairs of functions.
• B and E are linear transformations. Differentiation is linear, and adding or multiplying by a function (provided the function is not dependent on f(t)) does not affect linearity.
• C is not linear. Evaluating a function at a specific point does not preserve additivity or homogeneity.
• D is not linear because of the square term which violates both additivity and homogeneity.

Thus, the linear transformations among the options are B and E.