Question

If L: P4 + P3 is a function defined by L(p(t)= -3 p'(t)+2 p'(-4), then L is not a linear transformation Select one: True False

Answer 1

Upon verifying both additivity and homogeneity properties for the given function L, it is confirmed that L is a linear transformation.

Step-by-step explanation:

To determine if the given function L: P_4 to P_3 defined by L(p(t)) = -3 p'(t) + 2 p'(-4) is a linear transformation, we must verify if it satisfies two properties: additivity and homogeneity. These two properties are defined as:

• L(u + v) = L(u) + L(v) for all u, v in P_4
• L(a * v) = a * L(v) for all scalars a and all v in P_4

Lets test the first property, additivity:

L(p(t) + q(t)) = -3 (p+q)'(t) + 2 (p+q)'(-4)
= (-3 p'(t) + 2 p'(-4)) + (-3 q'(t) + 2 q'(-4))
= L(p(t)) + L(q(t))

Now, lets check the second property, homogeneity:

L(a * p(t)) = -3 (a*p(t))' + 2 (a*p(t))'(-4)
= -3 a * p'(t) + 2 a * p'(-4)
= a * L(p(t))

Since the function L satisfies both additivity and homogeneity, it is indeed a linear transformation.

Answer 2

L is a linear transformation because it satisfies both the additivity property and the homogeneity property required for linear transformations. This is demonstrated through examples applied to both properties using arbitrary polynomials and a scalar.

Step-by-step explanation:

The student asks whether L: P4 → P3 defined by L(p(t)) = -3 p'(t) + 2 p'(-4) is a linear transformation. To determine if L is a linear transformation, we need to check two main properties: the additivity property (L(u + v) = L(u) + L(v)) and the homogeneity property (L(cu) = cL(u)) for any polynomials u, v and any scalar c.

Let's first consider additivity. Suppose we have two polynomials p(t) and q(t). Then,

L(p(t) + q(t)) = -3(p + q)'(t) + 2(p + q)'(-4) = -3(p'(t) + q'(t)) + 2(p'(-4) + q'(-4)) = (-3p'(t) + 2p'(-4)) + (-3q'(t) + 2q'(-4)) = L(p(t)) + L(q(t)).

Now, let's check homogeneity. For any scalar c and a polynomial p(t),

L(c · p(t)) = -3(cp(t))' + 2(cp(t))'(-4) = c(-3p'(t)) + c(2p'(-4)) = c(-3p'(t) + 2p'(-4)) = cL(p(t)).

Since L satisfies both additivity and homogeneity, it is indeed a linear transformation.