# Prove whether {f}:mathbb{R}tomathbb{R} text{defined by} f{{left({x}right)}}={4}{x}-{2} is a linear transformation. Prove whether {f}:mathbb{R}tomathbb

Prove whether is a linear transformation.
Prove whether is a linear transformation.
Which one equivalent to the linear transformation
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Step 1
To prove that $f\left(x\right)$ is linear transformation we have to prove that
$f\left(x+y\right)=f\left(x\right)+f\left(y\right)$
And af $\left(x\right)=f\left(ax\right),$ a belongs to R
Here a is a scalar and R is the set of real numbers.
Step 2

Now we check if $f\left(x+y\right)=f\left(x\right)+f\left(y\right)$
$f\left(x+y\right)=4\left(x+y\right)-2=4x+4y-2$
And,
$f\left(x\right)+f\left(y\right)=4x-2+4y-2=4x+4y-4$

Hence $f\left(x\right)=4x-2$ is not a linear transformation.
Step 3

$f\left(x+y\right)=2\left(x+y\right)=2x+2y$
And,
$f\left(x\right)+f\left(y\right)=2x+2y$
We get, $f\left(x+y\right)=f\left(x\right)+f\left(y\right)$
And,
af $\left(x\right)=a\left(2x\right)=2ax=f\left(ax\right)$
Both the properties are satisied, hence $f\left(x\right)=2x$ is a linear transformation.
Step 4

Hence