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Q18E

Expert-verifiedFound in: Page 184

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism****, ${\mathit{T}}{\left(x+iy\right)}{\mathbf{=}}{{\mathit{x}}}^{{\mathbf{2}}}{\mathbf{+}}{{\mathit{y}}}^{{\mathbf{2}}}$** **from** ${\mathbf{\u2102}}$to ${\mathbf{\u2102}}$.

The transformation $T(\mathrm{x}+\mathrm{iy})={x}^{2}+{y}^{2}$ is not a linear transformation.

**Consider two linear spaces V and W. A transformation T is said to be a linear transformation if it satisfies the properties,**

${\mathit{T}}\mathbf{(}\mathbf{f}\mathbf{+}\mathbf{g}\mathbf{)}{\mathbf{=}}{\mathit{T}}{\mathbf{\left(}}{\mathit{f}}{\mathbf{\right)}}{\mathbf{+}}{\mathit{T}}{\mathbf{\left(}}{\mathit{g}}{\mathbf{\right)}}\phantom{\rule{0ex}{0ex}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathit{T}}{\mathbf{\left(}}{\mathit{k}}{\mathit{v}}{\mathbf{\right)}}{\mathbf{=}}{\mathit{k}}{\mathit{T}}{\mathbf{\left(}}{\mathit{v}}{\mathbf{\right)}}$

** **

**For all elements f,g of v and k is scalar.**

**An invertible linear transformation is called an isomorphism.**

Consider the transformation $T(\mathrm{x}+\mathrm{iy})={x}^{2}+{y}^{2}$ , from $\mathrm{\u2102}$to $\mathrm{\u2102}$.

Check whether the transformation satisfies the below two conditions or not.

$1.T(A+B)=T\left(A\right)+T\left(B\right)\phantom{\rule{0ex}{0ex}}2.T\left(kA\right)=kT\left(A\right)$

Verify the first condition.

Let $A={x}_{1}+i{y}_{1}$ and $B={x}_{2}+i{y}_{2}$ be arbitrary complex numbers from $\mathrm{\u2102}$. Then,

$T(A+B)=T({x}_{1}+i{y}_{1}+{x}_{2}+i{y}_{2})\phantom{\rule{0ex}{0ex}}=T\left(\left({x}_{1}+{x}_{2}\right)+i\left({y}_{1}+{y}_{2}\right)\right)\phantom{\rule{0ex}{0ex}}={\left({x}_{1}+{x}_{2}\right)}^{2}+{\left({y}_{1}+{y}_{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}={x}_{1}^{2}+{x}_{2}^{2}+2{x}_{1}{x}_{2}+{y}_{1}^{2}+{y}_{2}^{2}+2{y}_{1}{y}_{2}\phantom{\rule{0ex}{0ex}}=\left({x}_{1}^{2}+{y}_{1}^{2}\right)+\left({x}_{2}^{2}+{y}_{2}^{2}\right)+2{x}_{1}{x}_{2}+2{y}_{1}{y}_{2}\phantom{\rule{0ex}{0ex}}=T\left(A\right)+T\left(B\right)+2{x}_{1}{x}_{2}+2{y}_{1}{y}_{2}$

It is clear that, the first condition $T\left(A+B\right)\ne T\left(A\right)+T\left(B\right)$ is not satisfied.

Thus, *T* is not a linear transformation.

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