Question

Show that the transformation T defined by T(x₁, x₂) = (4x₁-2x₂, 3|x₂|) is not linear.

Answer 1

To show that the transformation T is not linear, we need to demonstrate that it does not satisfy the properties of linearity for addition and scalar multiplication. By evaluating the transformation for addition and scalar multiplication, we can see that it satisfies both properties, indicating that it is linear.

Step-by-step explanation:

To show that the transformation T defined by T(x₁, x₂) = (4x₁-2x₂, 3|x₂|) is not linear, we need to demonstrate that it does not satisfy the properties of linearity. A linear transformation should preserve addition and scalar multiplication.

Let's consider the transformation of addition:

T((x₁, x₂) + (y₁, y₂)) = T((x₁ + y₁, x₂ + y₂)) = (4(x₁ + y₁) - 2(x₂ + y₂), 3|x₂ + y₂|)

Expanding and simplifying, we have:

(4x₁ - 2x₂ + 4y₁ - 2y₂, 3|x₂ + y₂|)

Now let's consider the addition of individual transformations:

T(x₁, x₂) + T(y₁, y₂) = (4x₁ - 2x₂, 3|x₂|) + (4y₁ - 2y₂, 3|y₂|)

Expanding and simplifying:

(4x₁ - 2x₂ + 4y₁ - 2y₂, 3|x₂ + y₂|)

We can see that the two results are equal, which satisfies the property of linearity for addition. Now let's consider scalar multiplication:

T(c(x₁, x₂)) = T(cx₁, cx₂) = (4(cx₁) - 2(cx₂), 3|cx₂|)

Simplifying:

(4cx₁ - 2cx₂, 3|cx₂|)

Now let's consider the scalar multiplication of a transformed vector:

cT(x₁, x₂) = c(4x₁ - 2x₂, 3|x₂|) = (4cx₁ - 2cx₂, 3|cx₂|)

We can see that the two results are equal, which also satisfies the property of linearity for scalar multiplication.

Since the transformation T satisfies the properties of linearity for addition and scalar multiplication, we can conclude that it is linear.