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Which of the following transformations are linear?

Verify which of the following transformations are linear.

  • $T_1(x_1,x_2,x_3) = (x_1,0,x_3)$
  • $T_2(x_1,x_2)=(2x_1 – 3x_2,x_1 +4,5x_2)$
  • $T_3(x_1,x_2,x_3)=(1,x_2,x_3)$
  • $T_4(x_1,x_2)=(4x_1 – 2x_2,3|x_2|)$
  • $T_5(x_1,x_2,x_3)=(x_1,x_2,-x_3)$

The objective of this question is to find the linear transformation from the given transformation.

This question uses the concept of linear transformation. The linear transformation is the mapping of one vector space to another vector space that preserves the underlying structure and also preserves the arithmetic operations which are the multiplication and addition of vectors. A linear transformation is also called a Linear operator.

Expert Answer

For linear transformation, the following  criteria must be satisfied, which are:

$T(x+y)=T(x)+T(y)$

$T(ax)=a(Tx)$

$T(0)=0$

Where $a$ is a scalar.

a) To find if the given $T_1$ is a linear transformation or not, we have to satisfy the properties mentioned above of linear transformation.

So the given transformation is:

\[T_1(x_1,x_2,x_3)=(x_1,0,x_2)\]

\[T(x_1+y_1,x_2+y_2,x_3+y_3)=(x_1+y_1,0(x_2+y_2),x_3+y_3)\]

\[T(x_1,0,x_3)+T(y_1,0,y_2)\]

\[T(cx_1,cx_2,cx_3)=T(cx_1,(c)0,cx_3)\]

\[cT(x_1,0,x_3)\]

\[T(0,0,0)=0\]

So it is proved that the given transformation $T_1$ is a linear transformation.

b)  To find out if the given $T_2$ is a linear transformation or not, we have to satisfy the properties mentioned above of linear transformation.

The given transformation is:

\[T(x_1,x_2)=(2x_1-3x_2+4,5x_2)\]

\[T(x_1+y_1,x_2+y_2)=(2(x_1+y_1)-3(x_2+y_2),(x_1+y_1)+4,5(x_2+y_2))\]

\[=(2x_1+2y_1-3x_2-3y_2,x_1+y_1+4,5x_2+5y_2)\]

\[T(x_1,x_2)+T(y_1,y_2)=(2x_1-3x_2,x_1+4,5x_2)+(2y_1-3y_2,y_1+4,5y_2)\]

\[=2x_1-3x_2+2y_1-3y_2,x_1+y_1+8,5x_2+5y_2)\neq T(x_1+y_2,x_2+y_2)\]

Hence, it is proved that $T_2$ is not a linear transformation.

c) Let $T:R^3$ is defined as:

\[T(x_1,x_2,x_3)=(1,x_2,x_3)\]

To prove whether T is a linear transformation or not,

Let $(x_1,x_2,x_3),(y_1,y_2,y_3)$ belongs to $R^3$ and $a$, $b$ are any constant or scalar.

Then, we have:

\[T((x_1,x_2,x_3)+(y_1,y_2,y_3))=T(x_1+y_1,x_2+y_2,x_3+y_3\]

\[=(1,x_2+y_2,x_3+y_3)\]

\[T(x_1,x_2,x_3)+T(y_1+y_2+y_3)=(1,x_2,x_3)+(1,y_2,y_3)\]

\[=(2,x_2+y_2,x_3+y_3)\]

Then:

\[T((x_1,x_2,x_3)+(y_1,y_2,y_3)) \neq T(x_1,x_2,x_3)+(y_1,y_2,y_3) \]

It is proved that the given transformation is not linear transformation.

d) Let $T$:$R^2 \rightarrow R^2$ is defined as:

\[T(x_1,x_2)=4x_1-2x_2,3|x_2|\]

In order to prove whether T is linear transformation or not,

Let $(x_1,x_2),(y_1,y_2,)$ belongs to $R^2$.

\[(x_1+y_1,x_2+y_2)=(4(x_1+y_1)-2(x_2+y_2),3|x_2+y_2|\]

\[=(4x_1+4y_1-2x_2-2y_2,3|x_2+y_2|)\]

\[=(4x_1-2x_2)+(4y_1-2y_2),3|x_2+y_2|\]

Where $|a+b|$ is less or equal to $|a|+|b|$.

Therefore, the given transformation is not linear.

You can do the same procedure for the transformations $T_5$ to find whether it is a linear transformation or not.

Numeric Answer

By using the  concept of linear transformation, it is proved that the transformation $T_1$, which is defined as:

\[T(x_1,x_2,x_3)=(x_1,0,x_2)\]

is a linear transformation, while other transformations are not linear.

Example

Show that the given transformation $T$ is a linear transformation or not.

\[T \begin{bmatrix} x\\ y\\ z \end{bmatrix} =  \begin{bmatrix}
x+y\\ x-z \end{bmatrix}  for all \begin{bmatrix} x\\ y\\ z\end{bmatrix} \in  R^3\]

Let $\overrightarrow{x_1}$ is :

\[=\begin{bmatrix} x1\\ y_1\\ z _1\end{bmatrix} \]

and $\overrightarrow{x_2}$ is :

\[=\begin{bmatrix} x2\\ y_2\\ z _2\end{bmatrix} \]

Then:

\[T(k \overrightarrow{x_1}+p\overrightarrow{x_2})= T\Bigg\{ (k \begin{bmatrix} x1\\ y_1\\ z _1\end{bmatrix} +p\begin{bmatrix} x2\\ y_2\\ z _2\end{bmatrix} \Bigg\} \]

\[= T\Bigg\{ ( \begin{bmatrix} kx1\\ ky_1\\ kz _1\end{bmatrix} +\begin{bmatrix} px2\\ py_2\\ pz _2\end{bmatrix} \Bigg\} \]

\[= T\Bigg\{ ( \begin{bmatrix} kx1+px2\\ ky_1+py_2\\ kz _1 +pz _2\end{bmatrix} \]

\[= \Bigg\{ ( \begin{bmatrix} (kx1+px2) +( ky_1+py_2)\\ (kx _1 +px_2)-(kz _1 +pz_2)\end{bmatrix} \]

\[=k\begin{bmatrix} x1+y_1\\ x_1+z_1\end{bmatrix}+p \begin{bmatrix} x2+y_2\\ x_2-z_2\end{bmatrix}\]

\[=kT \overrightarrow{x_1}+pT \overrightarrow{x_2}\]

Therefore, it is proved that the given transformation $ T \begin{bmatrix} x\\ y\\ z \end{bmatrix} =  \begin{bmatrix}
x+y\\ x-z \end{bmatrix}  for all \begin{bmatrix} x\\ y\\ z\end{bmatrix} \in  R^3$

is a linear transformation.

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