show that the transformation defined by T(x1,x2)=(2x1-3x2,x1+4,5x2) is not linear ChatGPT

A transformation T: R^n -> R^m is linear if it satisfies the following two properties for all vectors x, y in R^n and all scalars c in R:

1. T(x + y) = T(x) + T(y) (Additivity)
2. T(cx) = cT(x) (Homogeneity of degree 1)

Let's check if the given transformation T(x1, x2) = (2x1 - 3x2, x1 + 4.5x2) satisfies these properties.

1. Additivity:

Let's take two arbitrary vectors (x1, x2) and (y1, y2). Then,

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

On the other hand, T(x1, x2) + T(y1, y2) = (2x1 - 3x2, x1 + 4.5x2) + (2y1 - 3y2, y1 + 4.5y2) = (2x1 + 2y1 - 3x2 - 3y2, x1 + y1 + 4.5x2 + 4.5y2)

Comparing these two results, we see that they are equal, so the transformation satisfies the additivity property.

2. Homogeneity of degree 1:

Let's take an arbitrary vector (x1, x2) and an arbitrary scalar c. Then,

T(c(x1, x2)) = T((cx1, cx2)) = (2cx1 - 3cx2, cx1 + 4.5cx2)

On the other hand, cT(x1, x2) = c(2x1 - 3x2, x1 + 4.5x2) = (2cx1 - 3cx2, cx1 + 4.5cx2)

Comparing these two results, we see that they are equal, so the transformation satisfies the homogeneity of degree 1 property.

Therefore, the transformation T(x1, x2) = (2x1 - 3x2, x1 + 4.5x2) is linear.