Bukti bahwa ( T(v_1, v_2) = (v_1 + v_2, v_1) ) adalah transformasi linier

Syarat transformasi linier: \begin{align*} (1) \quad & T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \ (2) \quad & T(c\mathbf{u}) = cT(\mathbf{u}) \end{align*}

Misalkan:
$\( \mathbf{u} = (u_1, u_2), \quad \mathbf{v} = (v_1, v_2), \quad c \in \mathbb{R} \)$

---

Periksa syarat (1):

\begin{align*} T(\mathbf{u} + \mathbf{v}) &= T((u_1 + v_1, u_2 + v_2)) \ &= ((u_1 + v_1) + (u_2 + v_2), u_1 + v_1) \ &= (u_1 + u_2 + v_1 + v_2, u_1 + v_1) \end{align*}

\begin{align*} T(\mathbf{u}) + T(\mathbf{v}) &= (u_1 + u_2, u_1) + (v_1 + v_2, v_1) \ &= (u_1 + u_2 + v_1 + v_2, u_1 + v_1) \end{align*}

\[ \Rightarrow T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \quad \text{(✓)} \]

---

Periksa syarat (2):

\begin{align*} T(c\mathbf{u}) &= T((cu_1, cu_2)) \ &= (cu_1 + cu_2, cu_1) \ &= c(u_1 + u_2, u_1) \ &= cT(\mathbf{u}) \quad \text{(✓)} \end{align*}

---

Kesimpulan:
( T ) adalah transformasi linier.