Let \( T: R^2 \rightarrow R^2 \) be a linear transformation such that \( T(x, y) = (3x + 4y, 2x + 3y) \). Prove that \( T \) is a linear transformation.
Solution
Step 1 ：Step 1 : To prove that a transformation \( T \) is linear, we need to show that the following two properties hold: \[T(u + v) = T(u) + T(v)\] and \[T(cu) = cT(u)\] where \( u \) and \( v \) are any vectors in the domain and \( c \) is any scalar.
Step 2 ：Step 2 : Let's choose two arbitrary vectors \( u = (x_1, y_1) \) and \( v = (x_2, y_2) \) in \( R^2 \). Then, \[T(u + v) = T((x_1 + x_2, y_1 + y_2)) = (3(x_1 + x_2) + 4(y_1 + y_2), 2(x_1 + x_2) + 3(y_1 + y_2))\] which simplifies to \[(3x_1 + 3x_2 + 4y_1 + 4y_2, 2x_1 + 2x_2 + 3y_1 + 3y_2) = (3x_1 + 4y_1, 2x_1 + 3y_1) + (3x_2 + 4y_2, 2x_2 + 3y_2) = T(u) + T(v)\]
Step 3 ：Step 3 : Now, let's choose an arbitrary vector \( u = (x, y) \) and an arbitrary scalar \( c \) in \( R \). Then, \[T(cu) = T((cx, cy)) = (3(cx) + 4(cy), 2(cx) + 3(cy))\] which simplifies to \[(3cx + 4cy, 2cx + 3cy) = c(3x + 4y, 2x + 3y) = cT(u)\]
Step 4 ：Step 4 : Therefore, the transformation \( T \) is linear because it satisfies both properties of linearity.
