Question

For each of the following, determine whether T is a linear transformation.

a) T(x) = 2x
b) T(x) = x²
c) T(x) = √x
d) T(x) = 3x + 4

Answer 1

To determine whether T is a linear transformation, we need to check if it satisfies the additive and homogeneous properties. T(x) = 2x and T(x) = 3x + 4 are linear transformations, while T(x) = x² and T(x) = √x are not linear transformations.Option A & D are the correct answers.

Step-by-step explanation:

To determine whether T is a linear transformation, we need to check if it satisfies the two properties of a linear transformation: additive and homogeneous.

1. a) T(x) = 2x: This is a linear transformation because it satisfies both properties. For any two vectors x and y, T(x+y) = 2(x+y) = 2x + 2y = T(x) + T(y), and for any scalar c, T(cx) = 2(cx) = c(2x) = cT(x).
2. b) T(x) = x²: This is not a linear transformation because it does not satisfy the homogeneous property. For example, T(2x) = (2x)² = 4x² ≠ 2T(x).
3. c) T(x) = √x: This is not a linear transformation because it does not satisfy the additive property. For example, T(1+2) = √3 ≠ √1 + √2 = T(1) + T(2).
4. d) T(x) = 3x + 4: This is a linear transformation because it satisfies both properties. For any two vectors x and y, T(x+y) = 3(x+y) + 4 = 3x + 3y + 4 = (3x + 4) + (3y + 4) = T(x) + T(y), and for any scalar c, T(cx) = 3(cx) + 4 = c(3x) + 4 = cT(x).