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Mathematics
Linear and Abstract Algebra
072 is Q(theta) a linear transformation from R^2 to itself.
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[QUOTE="HOI, post: 6784307, member: 567247"] Better is to understand what "linear transformation" means! ANY transformation that can be written as a matrix multiplication is linear! A transformation, L, on a vector space is "linear" if and only if L(u+ v)= Lu+ Lv, for any vectors u and v, and L(au)= aLu, for any vector u and scalar, a. Here if $u= \begin{pmatrix}x \\ y \end{pmatrix}$ and $v= \begin{pmatrix} a \\ b\end{pmatrix}$, $L(u+ v)= \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{pmatrix}\begin{pmatrix}x+ a \\ y+ b\end{pmatrix}= \begin{pmatrix}(x+ a)cos(\theta)- (y+ b)sin(\theta) \\ (x+ a)sin(\theta)+ (y+ b)cos(\theta)\end{pmatrix}$. While $Lu+ Lv= \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix}+ \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{pmatrix}\begin{pmatrix} a \\ b \end{pmatrix}= \begin{pmatrix} xcos(\theta)- ysin(\theta) \\ xsin(\theta)+ y cos(\theta)\end{pmatrix}+ \begin{pmatrix} acos(\theta)- bsin(\theta) \\ asin(\theta)+ bcos(\theta)\end{pmatrix}= \begin{pmatrix}(x+ a)cos(\theta)- (y+ b)sin(\theta) \\ (x+ a)sin(\theta)+ (y+ b)cos(\theta)\end{pmatrix}$. And $L(au)= \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{pmatrix}\begin{pmatri https://mathhelpboards.com/forums/-/create-thread [SIZE=7][B]072 is Q(theta) is a linear transformation from R^2 to itself.[/B][/SIZE] [LIST] [*]Thread starter[USER=55]karush[/USER] [*]Start datehttps://mathhelpboards.com/threads/072-is-q-theta-is-a-linear-transformation-from-r-2-to-itself.28497/ [/LIST] https://mathhelpboards.com/threads/072-is-q-theta-is-a-linear-transformation-from-r-2-to-itself.28497/#post-124753https://mathhelpboards.com/threads/072-is-q-theta-is-a-linear-transformation-from-r-2-to-itself.28497/watch https://mathhelpboards.com/threads/072-is-q-theta-is-a-linear-transformation-from-r-2-to-itself.28497/post-124753 [LIST] [*]Thread starter [*]New [*]https://mathhelpboards.com/posts/124753/bookmark [*]https://mathhelpboards.com/threads/072-is-q-theta-is-a-linear-transformation-from-r-2-to-itself.28497/post-124753 [/LIST] [USER=55]https://mathhelpboards.com/data/avatars/s/0/55.jpg?1588030209[/USER] [B][USER=55]karush[/USER] Well-known member[/B] Jan 31, 2012 2,838 if Q(θ)Q(θ) is [cosθsinθ−sinθcosθ][cosθ−sinθsinθcosθ] how is Q(θ)Q(θ) is a linear transformation from R^2 to itself. ok I really didn't know a proper answer to this question but presume we would need to look at the unit circle not sure if this helps[ATTACH=full]314057[/ATTACH] Last edited by a moderator: Today at 12:07 AM https://mathhelpboards.com/posts/124753/react?reaction_id=1 https://mathhelpboards.com/threads/072-is-q-theta-is-a-linear-transformation-from-r-2-to-itself.28497/reply?quote=124753 https://mathhelpboards.com/threads/072-is-q-theta-is-a-linear-transformation-from-r-2-to-itself.28497/reply?quote=124753 https://mathhelpboards.com/posts/124753/report x}ax \\ ay \end{pmatrix}= \begin{pmatrix}ax coz [/QUOTE]
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072 is Q(theta) a linear transformation from R^2 to itself.
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