Determine whether each statement below is true or false. Justify each answer. a. A linear transformation is a special type of function. A. False. A linear transformation is not a function because it maps one vector x to more than one vector T(x). B. True. A linear transformation is a function from set of real numbers R to set of real numbers R that assigns to each vector x in set of real numbers R a vector T(x) in set of real numbers R. C. False. A linear transformation is not a function because it maps more than one vector x to the same vector T(x). D. True. A linear transformation is a function from set of real numbers R Superscript n to set of real numbers R Superscript m that assigns to each vector x in set of real numbers R Superscript n a vector T(x) in set of real numbers R Superscript m.

In order to obtain the correct answer we need to recall the definition of function (or map) from one set to another. So, we say that a function (or map) is a way to associate a unique object to every element of a set. In a more mathematical formulation we say that [tex]f:A\rightarrow B[/tex] is a function if, for every element [tex]a\in A[/tex] there exists a unique element [tex]f(a)=b\in B[/tex].

We need to recall the definition of linear transformation too. So, we say that a map [tex]T: \mathbb{R}^n\rightarrow \mathbb{R}^m[/tex] is a linear transformation such that

[tex]T(u+v) = T(u) + T(v)[/tex], for all [tex]u,v\in\mathbb{R}^n[/tex],

[tex]T(\alpha u) = \alpha T(u), for every [tex]u\in\mathbb{R}^n[/tex] and every [tex]\alpha\in\mathbb{R}.

Then, from the definition of linear transformations we already know that they are functions. But, in the particular case of linear transformations, we have:

[tex]A=\mathbb{R}^n[/tex] and [tex]B=\mathbb{R}^m[/tex],

[tex]T=f[/tex].

Also, a linear transformation must satisfies the conditions stated in its definition.

Just notice that these are the explanation that comes with the option D.