Question

Let :ℝ→ℝ be defined by ()=−72. is a linear transformation?

Answer 1

Yes, the function ()=−72 is a linear transformation.

Step-by-step explanation:

A linear transformation is a function that preserves the operations of vector addition and scalar multiplication. To determine if the function ()=−72 is a linear transformation, we need to check if it satisfies the properties of linearity:

In this case, let's check if the function satisfies these properties:

1. Property of Vector Addition:

Let's pick two random vectors, a and b:

a = 2

b = -3

Now, let's calculate the function applied to the sum of these vectors:

()=−72=−72 = -144

Now, let's calculate the sum of the function applied to each vector:

()+()=−72+(-72)=-144

Since the function satisfies the property of vector addition, let's move onto the second property.

2. Property of Scalar Multiplication:

Let's pick a random vector, a:

a = 5

Now, let's calculate the function applied to the scalar multiplication of this vector:

()=−72=−360

Now, let's calculate the scalar multiplication of the function applied to the vector:

=−72()=-72*5=-360

Since the function satisfies both properties of linearity, we can conclude that it is a linear transformation.