Linear Transformation Image: R3 to R2 - Finding the Output Range

To determine the image of T, we need to find all possible outputs of T for all inputs from R3.

Let's consider an arbitrary vector in R3, (x1, x2, x3). Applying the transformation T, we have:

T(x1, x2, x3) = (x2 - 2x3, 3x1 + x3)

Since the outputs are in R2, we can rewrite this as a vector equation:

(x, y) = (x2 - 2x3, 3x1 + x3)

From this equation, we can isolate x2 and x3:

x2 = x + 2x3 x3 = (y - 3x1)

Substituting these values back into the equation, we have:

(x, y) = (x + 2x3, 3x1 + x3) = (x + 2*(y - 3x1), 3x1 + (y - 3x1)) = (x + 2y - 6x1, -2x1 + y)

Therefore, the image of T is the set of all vectors (x, y) in R2 that can be expressed as (x + 2y - 6x1, -2*x1 + y) for some x1, x2, x3 in R.

In other words, the image of T is the entire R2 plane.