Question

Which of the following transformations are linear? Select all of the linear transformations. There may be more than one correct answer. Be sure you can justify your answers.

a) f(x)=2x+3
b) f(x)=x​
c) f(x)=x²
d) f(x)=1/x

Answer 1

A linear transformation is a function that maps a vector space to itself while preserving vector addition and scalar multiplication. The linear transformations in the given options are a) f(x) = 2x + 3 and b) f(x) = x.

Step-by-step explanation:

A linear transformation is a function that maps a vector space to itself in a way that preserves the vector addition and scalar multiplication. In other words, a linear transformation satisfies two properties:

1. Additivity: f(x + y) = f(x) + f(y) for all vectors x and y in the vector space.
2. Scalar multiplication: f(cx) = cf(x) for all scalars c and vectors x.

Out of the options given, a) f(x) = 2x + 3 and b) f(x) = x are linear transformations because they satisfy both additivity and scalar multiplication. The other options, c) f(x) = x² and d) f(x) = 1/x, are not linear transformations because they do not satisfy the additivity property.