Question

Regarding linear algebra, and vector spaces etc, how do you know if a transformation is considered to be linear? (eg given in question)

Answer 1

T : M_22 → R3 where T \[\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]\] = [3 a−d, b−2 c, 3 c+3 d]

Answer 2

Would the above example be a linear transformation? Why/why not? Thanks!!

Answer 3

A transformation is considered linear if it satisfies the following properties.
\[T(u + v) =T(u) + T(v)\]\[T(av) =aT(v)\]

Answer 4

So closed under addition and scalar multiplication.... But can I ask about the example above? How would you put those rules into an example? :S

Answer 5

\[T(k(a_1, b_1, c_1, d_1) + (a_2, b_2, c_2, d_2))\]\[=T(ka_1 + a_2, kb_1 + b_2, kc_1 + c_2, kd_1 + d_2)\]\[=(3(ka_1 + a_2) - (kd_1 + d_2), (kb_1 + b_2) - 2(kc_1 + c_2), 3(kc_1 + c_2) + 3(kd_1 + d_2))\]\[=(3ka_1 + 3a_2 - kd_1 - d_2, kb_1 + b_2 - 2kc_1 - 2c_2, 3kc_1 + 3c_2 + 3kd_1 + 3d_2)\]\[=(k(3a_1 - d_1) + (3a_2 - d_2), k(b_1 - 2c_1) + (b_2 - 2c_2), k(3c_1 + 3d_1) + (3c_2 + 3d_2))\]\[=(k(3a_1 - d_1) , k(b_1 - 2c_1), k(3c_1 + 3d_1)) + (3a_2 - d_2, b_2 - 2c_2, 3c_2 + 3d_2)\]\[=k(3a_1 - d_1,b_1 - 2c_1,3c_1 + 3d_1) + (3a_2 - d_2, b_2 - 2c_2, 3c_2 + 3d_2)\]\[=kT(a_1, b_1, c_1, d_1) + T(a_2, b_2, c_2, d_2)\]
Therefore T is a linear transformation.

Answer 6

This is true since T satisfies the two properties I described above.

Answer 7

Alchemista's answer is correct. Both properties were shown at once in the example but it could have been split up for clarity for you own interests. This is the general process to test for linearity.