Short Answer

Which of the (nonlinear) transformations from $R^{2} to R^{2}$ to in Exercises 25 through 27 are invertible? Find the inverse if it exists.26. role="math" localid="1660365558437" $[\begin{matrix} y_{1} \\ y_{2} \end{matrix}] = [\begin{matrix} x_{2} \\ x_{1}^{3} + x_{2} \end{matrix}]$

The inverse of the given transformation is $|\begin{matrix} x_{1} = \sqrt[3]{y_{2} - y_{1}} \\ x_{2} = y_{1} \end{matrix}|$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Check whether the transformation is invertible or not

The given transformation is $|\begin{matrix} y_{1} = x_{2} \\ y_{2} = x_{1}^{3} + x_{2} \end{matrix}|$ .

We can rewrite the transformation as $f; R^{2} \to R^{2}$ .such that $f (x_{1}, x_{2}) = (x_{2}, x_{1}^{3} + x_{2})$

Then the given transformation is invertible if and only if f is invertible.

(a) Suppose $f (x_{1}, x_{2}) = f (z_{1}, z_{2})$ for some $(x_{1}, x_{2}), f (z_{1}, z_{2}) \in R^{2}$ .Then $x_{1}^{3} = z_{1}^{3}$ and .So $x_{1} z_{1}$ are zero or nonzero together.

Now $(x_{1} - z_{1}) (x_{1}^{2} + x_{1} z_{1} + z_{1}^{2}) = x_{1}^{3} - z_{1}^{3} = 0$ (1)

Now $x_{1}^{2} + x_{1} z_{1} + z_{1}^{2} = 0 x_{1} z_{1} = {(x_{1} + z_{1})}^{2}$ and $- 3 x_{1} z_{1} = {(x_{1} - z_{1})}^{2}$ . So $x_{1}, z_{1} = 0$ . If $x_{1} z_{1}$ are nonzero, then ${x_{1}}^{2} + x_{1} z_{1} + {z_{1}}^{2} = 0$ and hence from (1) $x_{1} = z_{1}$ .If $x_{1}, z_{1}$ are zero, then $x_{1} = z_{1}$ .Thus $x_{1} = z_{1}$ and ${x_{1}}^{3} = {z_{1}}^{3}$ . Sofis one-one.

(b) Let $(z_{1}, z_{2}) \in R^{2}$ . Then there exists $x_{1} \in R$ such that ${x_{1}}^{3} = z_{2} - z_{1}$ . Therefore $f (x_{1}, z_{1}) = (z_{1}, {x_{1}}^{3} + z_{1}) = (z_{1}, z_{2})$ . Hence fis onto.

Step 2: Find the inverse of the given transformation

So,fis invertible and therefore the given transformation is invertible. Now $f^{- 1} (z_{1}, z_{2}) = (\sqrt[3]{z_{2} - z_{1}}, z_{1})$ Therefore inverse of the given transformation is $|\begin{matrix} x_{1} = \sqrt[3]{y_{2} - y_{1}} \\ x_{2} = y_{1} \end{matrix}|$

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Short Answer

Which of the (nonlinear) transformations from $R^{2} to R^{2}$ to in Exercises 25 through 27 are invertible? Find the inverse if it exists.26. role="math" localid="1660365558437" $[\begin{matrix} y_{1} \\ y_{2} \end{matrix}] = [\begin{matrix} x_{2} \\ x_{1}^{3} + x_{2} \end{matrix}]$

Step by Step Solution

Step 1: Check whether the transformation is invertible or not

Step 2: Find the inverse of the given transformation

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Linear Algebra With Applications

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Which of the (nonlinear) transformations from R2 to R2to in Exercises 25 through 27 are invertible? Find the inverse if it exists.26. role="math" localid="1660365558437" [y1y2]=[x2x13+x2]

Step by Step Solution

Step 1: Check whether the transformation is invertible or not

Step 2: Find the inverse of the given transformation

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Which of the (nonlinear) transformations from $R^{2} to R^{2}$ to in Exercises 25 through 27 are invertible? Find the inverse if it exists.26. role="math" localid="1660365558437" $[\begin{matrix} y_{1} \\ y_{2} \end{matrix}] = [\begin{matrix} x_{2} \\ x_{1}^{3} + x_{2} \end{matrix}]$