Question

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.

Answer 1

The correct option are A and D

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

`T(a^2 x) = (T(ax))^2`

Comparing the RHS and the LHS of the equation we see that the independent variable has been modified from
`x \to x^2` 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