Given the transformation L : R3 rightarrow R be defined by L [x y z] = x + 2y + 3z. show that L is linear. Solution to show that L is linear we will show that L(aX1+bX2) = aL(X1) + bL(X2) Where X1 = (x1 y1 z1) => aX1 = (ax1 ay1 az1) and X2 = (x2 y2 z2) => bX2 = (bx2 by2 bz2) => aX1 + bX2 = (ax1+bx2 ay1+by2 az1+bz2) L( aX1+bX2) = ax1+bx2 + 2(ay1+by2) + 3(az1+bz2) =a(x1+ 2y1 + 3z1) + b(x2+ 2y2 + 3z2) =aL(X1) + bL(X2) Hence Proved L is linear as it satisfies both closed under vector addition and scalar multiplication.