Answer to Question #194951 in Linear Algebra for Muhammad Haziq

Question #194951

Determine whether the function T: R

3 →R

3 by 𝑇 ([

𝑥

𝑦

𝑧

]) = [

𝑥 + 𝑦 − 𝑧

2𝑥𝑦

𝑥 + 𝑧 + 1

] is a linear 

transformation between vector space.



1
Expert's answer
2021-05-19T17:17:35-0400

The given function-


T([xyz])=[x+yz2xyx+z+1]T(\begin{bmatrix} x\\y\\z \end{bmatrix})=\begin{bmatrix}x+y-z\\2xy\\x+z+1\end{bmatrix}


    Letα=[α1α2α3],β=[β1β2β3]R3\alpha=\begin{bmatrix} \alpha_1\\\alpha_2\\\alpha_3 \end{bmatrix},\beta=\begin{bmatrix} \beta_1\\\beta_2\\\beta_3 \end{bmatrix}\in R^3


T(α+β)=T([α1α2α3]+[β1β2β3])=T([α1+β1α2+β2α3+β3])=[α1+β1+α2+β2(α3+β3)2(α1+β1)(α2+β2)α1+β1+α3+β3+1]=[α1+α2α32α1α2α1+α3+1]+[β1+β2β32β1β2+2(α1β2+β1α2)β1+β3]T(α)+T(β)T(\alpha+\beta)=T(\begin{bmatrix} \alpha_1\\\alpha_2\\\alpha_3 \end{bmatrix}+\begin{bmatrix} \beta_1\\\beta_2\\\beta_3 \end{bmatrix}) \\[9pt] =T(\begin{bmatrix} \alpha_1+\beta_1\\\alpha_2+\beta_2\\\alpha_3+\beta_3 \end{bmatrix}) \\[9pt] =\begin{bmatrix} \alpha_1+\beta_1+\alpha_2+\beta_2-(\alpha_3+\beta_3)\\2(\alpha_1+\beta_1)(\alpha_2+\beta_2)\\\alpha_1+\beta_1+\alpha_3+\beta_3+1 \end{bmatrix}\\[9pt] =\begin{bmatrix} \alpha_1+\alpha_2-\alpha_3\\2\alpha_1 \alpha_2\\\alpha_1+\alpha_3+1 \end{bmatrix}+\begin{bmatrix} \beta_1+\beta_2-\beta_3\\2\beta_1 \beta_2+2(\alpha_1 \beta_2+\beta_1 \alpha_2)\\\beta_1+\beta_3 \end{bmatrix}\\[9pt]\neq T(\alpha)+T(\beta)




Since, T(β)=[α1+α2α32α1alpha2α1+alpha3+1][β1+β2β32β1β2+2(α1β2+β1α2)β1+β3]T(\beta)=\begin{bmatrix} \alpha_1+\alpha_2-\alpha_3\\2\alpha_1 alpha_2\\\alpha_1+alpha_3+1 \end{bmatrix}\neq\begin{bmatrix} \beta_1+\beta_2-\beta_3\\2\beta_1 \beta_2+2(\alpha_1 \beta_2+\beta_1 \alpha_2)\\\beta_1+\beta_3 \end{bmatrix}


Therefore T is not liner transformation between vector space.

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