Haven

2020-12-15

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

Prove whether$f:\mathbb{R}\to \mathbb{R}\text{}\text{defined by}\text{}f\left(x\right)=2x$ is a linear transformation.

Which one equivalent to the linear transformation$T:\mathbb{R}\to \mathbb{R}\text{}\text{defined by}\text{}T\left(1\right)=2?$

Prove whether

Which one equivalent to the linear transformation

SchepperJ

Skilled2020-12-16Added 96 answers

Step 1

To prove that$f(x)$ is linear transformation we have to prove that

$f(x+y)=f\left(x\right)+f\left(y\right)$

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

$f\left(x\right)=4x-2\text{}\text{and let}\text{}f\left(y\right)=4y-2$

Now we check if$f(x+y)=f\left(x\right)+f\left(y\right)$

$f(x+y)=4(x+y)-2=4x+4y-2$

And,

$f\left(x\right)+f\left(y\right)=4x-2+4y-2=4x+4y-4$

$\Rightarrow f(x+y)\text{}\text{is not equal to}\text{}f\left(x\right)+f\left(y\right)$

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

Step 3

$f\left(x\right)=2x\text{}\text{and}\text{}{\displaystyle f\left(y\right)=2y}$

$f(x+y)=2(x+y)=2x+2y$

And,

$f\left(x\right)+f\left(y\right)=2x+2y$

We get,$f(x+y)=f\left(x\right)+f\left(y\right)$

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

$f\left(x\right)=2x\text{}at\text{}x=1,f\left(1\right)=2$

Hence$f:R\to R,f\left(x\right)=2x\text{}\text{is equivalent to}\text{}{\displaystyle T:R\to R,T\left(1\right)=2.}$

To prove that

And af

Here a is a scalar and R is the set of real numbers.

Step 2

Now we check if

And,

Hence

Step 3

And,

We get,

And,

af

Both the properties are satisied, hence

Step 4

Hence

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