不行

歐式空間是線性空間，所有矩陣乘法標量乘法是線性變換，原命題即求證 f((a, b, c), (x, y, z)) = a*x, b*y, c*z 是否為線性變換。

f((a1+a2, b1+b2, c1+c2), (x1+x2, y1+y2, z1+z2)) = (a1+a2)*(x1+x2), (b1+b2)*(y1+y2), (c1+c2)*(z1+z2)

f((a1, b1, c1), (x1, y1, z1)) + f((a2, b2, c2), (x2, y2, z2)) = a1*x1+a2*x2, b1*y1+b2*y2, c1*z1+c2*z2

兩式不相等，即f不為線性變換。

--- 正式證明 ---

Let be the vector space of , be the vector space of . Define the function by: . Prove or disprove is a linear operator.

Let , , .

Then ,

.

So is not a linear operator.