Identify all “linear functions” that are also “linear transformations” from $\mathbb R$ to $\mathbb R$ as defined in Linear Algebra.

Question

Full Question: $\mathbb R$ is a linear space. In algebra, a function $f$ of the form $f(x) = mx + b$ is called a “linear function”. E.g. $f(x) = 2x+ 1$. Identify all “linear functions” that are also “linear transformations” from $\mathbb R$ to $\mathbb R$ as defined in Linear Algebra.

I am having trouble answering this question. Is it asking to find all the linear functions that are also linear transformations for the equation $f(x) = 2x + 1$?

Please help. I really want to understand this concept.

Answer 1

They are noting that any function of the form $f(x) = mx + b$ is a linear function. However, it is not true that all of these linear functions are also linear transformations from $\mathbb{R} \to \mathbb{R}$. (Why?). What they are asking is for you to identify which linear functions of the form $f(x) = mx + b$ are also linear transformations.

The specific example of $f(x) = 2x+1$ is an example of a linear function that is not a linear transformation. If you can figure out why this is, you should be able to answer the question in the general sense.

Answer 2

A linear transformation has the property that $f(x + y) = f(x) + f(y)$

$f(x) = mx + b$

If $f$ is a linear transformation then:

$f(x+y)= m(x+y) + b = (mx+b) + (my+b)$

For what values of $m,b$ is this true?

Answer 3

Suppose $$f:\mathbb R\to \mathbb R $$is a linear transformation.

Let $$f(1)=a$$

For an arbitrary $x\in \mathbb R$, we have $$f(x) =f(x\times 1)=x\times f(1) = ax$$

Thus the only linear transformations are functions defined as $f(x)=ax$