A matrix A acts on a vector X by multiplication to result in

A matrix A \"acts\" on a vector X by multiplication to result in AX. What would it mean to say that this \"action\" is linear? Is it true that thus action is linear? Prove or find a counterexample.

Solution

We have been given a Matrix A and we are multiplying it to a vector X to get AX. By calling this \"action of multiplication\" linear, we would actual mean that matrix A is a Linear Transformation.

Yes this action is indeed a linear. In order to prove this action as linear (or in other words, to show that A is a linear transformation, we need to show that it is closed under vector addition and scalar multiplication.

Closeness under vector addition:;

Say there be two matrices A1 and A2. Therefore,

(A1+A2)X = A1X+A2X is true by distributive property of matrices. Therefore, the given multiplication action is closed under vector addition.

Closeness under scalar multiplication:

A(kX) = k(AX) is also true, therefore, the given multiplication action is closed under scalar multiplication.

Hence, the given action is linear.