Science:Math Exam Resources/Courses/MATH221/December 2007/Question Section 103 10 (a)
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Question Section 103 10 (a) |
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A discrete dynamical system is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{aligned}x_{n+1}&=x_{n}+3y_{n}\quad \quad x_{0}=5\\y_{n+1}&=2x_{n}+2y_{n}\quad \quad y_{0}=10\end{aligned}}} Find explicit formulas for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle x_{n}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle y_{n}} , with the given initial condition. |
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Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? |
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! |
Hint |
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Science:Math Exam Resources/Courses/MATH221/December 2007/Question Section 103 10 (a)/Hint 1 |
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Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
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Solution |
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Let us first write the system in matrix notation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}x_{n+1}\\y_{n+1}\end{pmatrix}}={\begin{pmatrix}1&3\\2&2\end{pmatrix}}{\begin{pmatrix}x_{n}\\y_{n}\end{pmatrix}}.} Notice that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}={\begin{pmatrix}1&3\\2&2\end{pmatrix}}{\begin{pmatrix}x_{0}\\y_{0}\end{pmatrix}},} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}x_{2}\\y_{2}\end{pmatrix}}={\begin{pmatrix}1&3\\2&2\end{pmatrix}}^{2}{\begin{pmatrix}x_{0}\\y_{0}\end{pmatrix}},} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}x_{3}\\y_{3}\end{pmatrix}}={\begin{pmatrix}1&3\\2&2\end{pmatrix}}^{3}{\begin{pmatrix}x_{0}\\y_{0}\end{pmatrix}}...} Therefore, we hypothesize that for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle n\in \mathbb {N} ,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}x_{n}\\y_{n}\end{pmatrix}}={\begin{pmatrix}1&3\\2&2\end{pmatrix}}^{n}{\begin{pmatrix}x_{0}\\y_{0}\end{pmatrix}}.} This explicit formula can (and should) be established rigorously by induction (we leave this as an exercise). Now we want to compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}1&3\\2&2\end{pmatrix}}^{n}} . Recall that matrix powers are easy to compute via matrix diagonalization. Therefore, we find the eigenvalues and eigenvectors for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle A={\begin{pmatrix}1&3\\2&2\end{pmatrix}}} . The eigenvalues are determined from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )-6=(\lambda -4)(\lambda +1)=0} , so the eigenvalues of A are 4 and -1. To find an eigenvector associated with the eigenvalue 4, we solve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle (A-4I)x=0} , i.e. the equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}-3&3\\2&-2\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}},} from which it is clear that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}1\\1\end{pmatrix}}} is an eigenvector. To find an eigenvector associated with the eigenvalue -1, we solve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}2&3\\2&3\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}},} from which we find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}-3\\2\end{pmatrix}}} is an eigenvector. Now we form the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle P={\begin{pmatrix}1&-3\\1&2\end{pmatrix}}} whose columns are the eigenvectors of A, and the diagonal matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \Lambda ={\begin{pmatrix}4&0\\0&-1\end{pmatrix}}} with the corresponding eigenvalues. We know that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle A=P\Lambda P^{-1}} and hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle A^{n}=P\Lambda ^{n}P^{-1}=P{\begin{pmatrix}4^{n}&0\\0&(-1)^{n}\end{pmatrix}}P^{-1}.} The final step is to compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle P^{-1}={\frac {1}{5}}{\begin{pmatrix}2&3\\-1&1\end{pmatrix}}} , and then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle A^{n}} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle A^{n}={\frac {1}{5}}{\begin{pmatrix}1&-3\\1&2\end{pmatrix}}{\begin{pmatrix}4^{n}&0\\0&(-1)^{n}\end{pmatrix}}{\begin{pmatrix}2&3\\-1&1\end{pmatrix}}={\frac {4^{n}}{5}}{\begin{pmatrix}2&3\\2&3\end{pmatrix}}+{\frac {(-1)^{n}}{5}}{\begin{pmatrix}3&-3\\-2&2\end{pmatrix}}.} Check that for n = 1 we recover the matrix A. Finally, the question asks us to find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle x_{n}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle y_{n}} explicitly, with the given initial condition. Having computed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle A^{n}} , we can now compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{pmatrix}x_{n}\\y_{n}\end{pmatrix}}={\begin{pmatrix}1&3\\2&2\end{pmatrix}}^{n}{\begin{pmatrix}x_{0}\\y_{0}\end{pmatrix}}=\left[{\frac {4^{n}}{5}}{\begin{pmatrix}2&3\\2&3\end{pmatrix}}+{\frac {(-1)^{n}}{5}}{\begin{pmatrix}3&-3\\-2&2\end{pmatrix}}\right]{\begin{pmatrix}5\\10\end{pmatrix}}.} Upon multiplying and simplifying the above equation, we find that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle {\begin{aligned}x_{n}&=2(4)^{n+1}+3(-1)^{n+1}\\y_{n}&=2[4^{n+1}+(-1)^{n}]\end{aligned}}} . |
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