Question

Show that the transformation T defined by T(x, y) = (2x - 3y, x+4.5y) is not linear.

I believe it satisfies both Additivity and Homogeneity, so I think it is a linear transformation.

Answer 1

Shown below

Explanation:

Let's check whether the transformation T satisfies both Additivity and Homogeneity.

Additivity:

For any two vectors (x1, y1) and (x2, y2), the following must hold:

T((x1, y1) + (x2, y2)) = T(x1, y1) + T(x2, y2)

Let's expand both sides of the equation:

T((x1, y1) + (x2, y2)) = (2(x1 + x2) - 3(y1 + y2), x1 + 4.5(y1 + y2))

T(x1, y1) + T(x2, y2) = (2x1 - 3y1, x1 + 4.5y1) + (2x2 - 3y2, x2 + 4.5y2) = (2x1 - 3y1 + 2x2 - 3y2, x1 + 4.5y1 + x2 + 4.5y2)

We can see that the two sides of the equation are not equal, so the transformation T is not additive.

Homogeneity:

For any scalar c and any vector (x, y), the following must hold:

T(c(x, y)) = cT(x, y)

Let's expand both sides of the equation:

T(c(x, y)) = (2(cx) - 3(cy), cx + 4.5(cy))

c T(x, y) = c(2x - 3y, x + 4.5y) = (2cx - 3cy, cx + 4.5cy)

We can see that the two sides of the equation are equal, so the transformation T is homogeneous.

Since the transformation T is not additive, it cannot be a linear transformation.