Prove that T(x,y)=(3x+y,2y,x-y) defines a linear transformation T:R^2 -> R^3.

Question

Prove that $T (x, y) = (3 x + y, 2 y, x - y)$ defines a linear transformation $T : R^{2} \to R^{3}$ . Give the full and correct answer.

Answer 1

Let

(x_{1}, y_{1}), (x_{2}, x_{2}) \in R^{2} and \leq t c_{1}, c_{2} \in R

. To prove that T is a linear transformation, we must prove that

T (c_{1} (x_{1}, y_{1}) + c_{2} (x_{2}, y_{2})) = c_{1} T (x_{1}, y_{1}) + c_{2} T (x_{2}, y_{2})

This is done by a direct computation:

T (c_{1} (x_{1}, y_{1}) + c_{2} (x_{2}, y_{2}))

= T (c_{1} x_{1}, c_{2} x_{2}) + (c_{2} x_{2}, c_{2} x_{2})

= T (c_{1} x_{1} c_{2} x_{2}, c_{1} y_{1} + c_{2} y_{2})

= (3 (c_{1} x_{1} + c_{2} x_{2}) + (c_{1} y_{1} + c_{2} y_{2}), 2 (c_{1} y_{1} + c_{2} y_{2}), (c_{1} x_{1} + c_{2} x_{2}) - (c_{1} y_{1} + c_{2} y_{2}))

= (3 c_{1} x_{1} + 3 c_{2} x_{2} + c_{1} y_{1} + c_{2} y_{2}, 2 c_{1} y_{1} + c_{2} y_{2}, c_{1} x_{1} + c_{2} x_{2} - c_{1} y_{1} + c_{2} y_{2})

= (3 c_{1} x_{1} + c_{1} y_{1}, 2 c_{1} y_{}

Expert Answer