Question

Which of the following defines a linear transformation from R^3 to R^2?

a. T(x, y, z) = (x + y, 2y - z)
b. T(x, y, z) = (2x - y, z)
c. T(x, y, z) = (x^2, yz)
d. T(x, y, z) = (xy, z)

Answer 1

The equation T(x, y, z) = (x + y, 2y - z) defines a linear transformation from R^3 to R^2.

Step-by-step explanation:

The equation T(x, y, z) = (x + y, 2y - z) defines a linear transformation from R^3 to R^2. To determine if a transformation is linear, we need to check two properties: additivity and scalar multiplication. In this case, let's consider the additivity property. Let's choose two arbitrary vectors (x1, y1, z1) and (x2, y2, z2) in R^3 and compute T(x1, y1, z1) + T(x2, y2, z2).

T(x1, y1, z1) + T(x2, y2, z2) = (x1 + y1, 2y1 - z1) + (x2 + y2, 2y2 - z2) = (x1 + x2 + y1 + y2, 2y1 + 2y2 - z1 - z2).

Now let's compute T(x1 + x2, y1 + y2, z1 + z2).

T(x1 + x2, y1 + y2, z1 + z2) = ((x1 + x2) + (y1 + y2), 2(y1 + y2) - (z1 + z2)) = (x1 + x2 + y1 + y2, 2y1 + 2y2 - z1 - z2).

Since T(x1, y1, z1) + T(x2, y2, z2) = T(x1 + x2, y1 + y2, z1 + z2), we can conclude that the transformation is linear.