Let T be a linear transformation, let the vectors x and y be in the domain of T, and let a and b be scalars. Select the statements that are ALWAYS TRUE. A. T(ax - by) = aT(x) – bT(y)
B. T (a²x) = (T(ax))2
C. T(x y) = T(x)T(y)
D. T(0) = 0
E. IfT(z) = 0, then z = 0.

The correct option are  A and D

Step-by-step explanation:

Given that T is  a linear transformation it means the T can only modify the value of the scalars but can not modify the dependent(y) and the independent variables (x)

Considering the first option

    T(ax -by)

evaluating the above we get

   Tax - Tbx

=>   a T(x) - bT(y)

So we see that in this equation T only just modified the scalars so option A will always be true

   Considering option B

Comparing the RHS and the LHS of the equation we see that the independent variable has been modified from hence option B is  false

 Considering option C

     T(x y) = T(x)T(y)

  Comparing the RHS and the LHS of the equation we see that the

The T modified both the independent variable and the independent variable hence C is  false

        Considering option D

    Comparing the RHS and the LHS of the equation we see that the

That there was no modification to the dependent and the independent variable  hence  

  option D is always  true

       Considering option E

      Comparing the RHS and the LHS of the equation we see that the

  For  RHS to be  true  z must not necessary be zero Hence option E is  false  

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step-by-step explanation:

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