A function T: $T : \Bbb{R^{1}} \to \Bbb{R^{1}} $ is a linear transform if and only if it is written on the form $T(x)=ax+b$?

Question

How do I determine if the following statement is true or false?

A function $T : \Bbb{R^{1}} \to \Bbb{R^{1}} $ is a linear transformation if and only if it can be written on the form $T(x)=ax+b$ where $a,b$ are constants.

Answer 1

In my definition a map $T \colon V \rightarrow W$ between two vector spaces $V,W$ over a field $\mathbb{K}$ is linear if $T(x+y)=T(x)+T(y)$ and $T(\lambda x)= \lambda T(x)$ for any $x,y \in V$ and $\lambda \in \mathbb{K}$.

Thus in your case to determine $T(x)$ observe that $x= x \cdot 1$ hence by linearity $T(x)=xT(1)$. You have to set $a=T(1)$ to get the equality $T(x)=ax$. The term $b$ must be zero, in case $b \neq 0$ the map $T$ is called an affine transformation.

Answer 2

What conditions does $T $ need to satisfy to be linear? What is $T (x+y)$? Is it equal to $T (x)+T (y)$?

Answer 3

What is your definition of linear transformation? Assuming that by linear transformation you mean a function that is closed under addition and under multiplication by a scalar your statement is clearly false. For example take $T(x)=x+1$. Then for all $x,y\in\mathbb{R}$ we have $T(x+y)=x+y+1$ but $T(x)+T(y)=x+1+y+1=x+y+2$.

The true statement is that $T:\mathbb{R}\to\mathbb{R}$ is a linear transformation if and only if it has the form $T(x)=ax$, which means $b$ must be zero.