# Given V = R, and two bases: B and C and the coordinator [v]B a) Find the change of coordinates matrix PB rightarrow C. b) Find the coordinator [v]C.

Question
Alternate coordinate systems
Given V = R, and two bases: B and C and the coordinator [v]B a) Find the change of coordinates matrix $$\displaystyle{P}{B}\rightarrow{C}.$$ b) Find the coordinator [v]C.

2021-02-20

a .$$\begin{bmatrix} 9 \\ 2 \end{bmatrix} = \prec_1 \begin{bmatrix} 2 \\ 1 \end{bmatrix} + \prec_2 \begin{bmatrix} -3 \\ 1 \end{bmatrix} \Rightarrow 2 \prec_1 + (-3) \prec_2 = 9 \prec_1 + \prec_2 = 2 \Rightarrow 2 \prec_1 + 2 \prec_2 = 4 2 \prec_1 + -3 \prec_2 = 9 \prec_2 = -1 \Rightarrow \prec_1 = 3 \begin{bmatrix} 4 \\ -3 \end{bmatrix} = B1 \begin{bmatrix} 2 \\ 1 \end{bmatrix} + B2 \begin{bmatrix} -3 \\ 1 \end{bmatrix} \Rightarrow 2 B1 - 3 B2 = 4 B1 + B2 = -3 \Rightarrow 2 B1 + 2 B2 = -6 2 B1 - 3 B2 = 4 \Rightarrow B2 = -2, B1 = -1 P_{B \leftarrow C} = \begin{bmatrix} \prec_1 & B1 \\ \prec_2 & B2 \end{bmatrix} = \begin{bmatrix} 3 & -1 \\ -1 & -2 \end{bmatrix}$$

b. $$​ Now \ v = \begin{bmatrix} 34 \\ 27 \end{bmatrix} = \curlyvee _1 \begin{bmatrix} 2 \\ 1 \end{bmatrix} \curlyvee_2 \begin{bmatrix} -3 \\ 1 \end{bmatrix} \Rightarrow 2 \curlyvee_1 - 3 \curlyvee_2 = 34 \curlyvee_1 + \curlyvee_2 = 27 \Rightarrow 2 \curlyvee_1 + 2 \curlyvee_2 = 54 \Rightarrow - 5 \curlyvee_2 = -20 \Rightarrow \curlyvee_2 = 4 \Rightarrow \curlyvee_1 = 23 \Rightarrow [v]C = \begin{bmatrix} \curlyvee_1 \\ \curlyvee_2 \end{bmatrix} = \begin{bmatrix} 23 \\ 4 \end{bmatrix}$$

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$$B_2 = \left\{ \left(\begin{array}{c}3\\ 1\end{array}\right),\left(\begin{array}{c}7\\ 2\end{array}\right) \right\}$$
find the change of basis matrix from $$\displaystyle{B}_{{1}}\to{B}_{{2}}$$ and next use this matrix to covert the coordinate vector
$$\overrightarrow{v}_{B_1} = \left(\begin{array}{c}2\\ -1\end{array}\right)$$ of v to its coodirnate vector
$$\overrightarrow{v}_{B_2}$$

To determine:
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$$\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ +\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}{\quad\text{and}\quad}{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ -\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ -\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}$$. Futhermore, it is a center of the corresponding linear system.
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c) To prove: $$\displaystyle{\left[{\frac{{{d}{r}}}{{{\left.{d}{t}\right.}}}}\ {<}\ {0}\right]}{\quad\text{and}\quad}{\left[{r}\rightarrow\ {0}\ {a}{s}\ {t}\rightarrow\ \infty\right]},$$ hence the critical point for the system $$\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ -\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ -\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}$$ is asymptotically stable and the solution of the initial value problem for $$\displaystyle{\left[{r}\ {w}{i}{t}{h}\ {r}={r}_{{{0}}}\ {a}{t}\ {t}={0}\right]}$$ becomes unbounded as $$\displaystyle{\left[{t}\rightarrow{\frac{{{1}}}{{{2}}}}\ {r}{\frac{{{2}}}{{{0}}}}\right]}$$, hence the critical point for the system $$\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ +\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}$$ is unstable.
a) Whether the statement, " The point with Cartesian coordinates $$\displaystyle{\left[\begin{array}{cc} -{2}&\ {2}\end{array}\right]}$$ has polar coordinates $$\displaystyle{\left[{b}{f}{\left({2}\sqrt{{{2}}},\ {\frac{{{3}\pi}}{{{4}}}}\right)}\ {\left({2}\sqrt{{{2}}},{\frac{{{11}\pi}}{{{4}}}}\right)}\ {\left({2}\sqrt{{{2}}},\ -{\frac{{{5}\pi}}{{{4}}}}\right)}\ {\quad\text{and}\quad}\ {\left(-{2}\sqrt{{2}},\ -{\frac{{\pi}}{{{4}}}}\right)}\right]}$$ " is true or false.
b) Whether the statement, " the graphs of $$\displaystyle{\left[{r}{\cos{\theta}}={4}\ {\quad\text{and}\quad}\ {r}{\sin{\theta}}=\ -{2}\right]}$$ intersect exactly once " is true or false.
c) Whether the statement, " the graphs of $$\displaystyle{\left[{r}={4}\ {\quad\text{and}\quad}\ \theta={\frac{{\pi}}{{{4}}}}\right]}$$ intersect exactly once ", is true or false.
d) Whether the statement, " the point $$\displaystyle{\left[\begin{array}{cc} {3}&{\frac{{\pi}}{{{2}}}}\end{array}\right]}{l}{i}{e}{s}{o}{n}{t}{h}{e}{g}{r}{a}{p}{h}{o}{f}{\left[{r}={3}{\cos{\ }}{2}\ \theta\right]}$$ " is true or false.
e) Whether the statement, " the graphs of $$\displaystyle{\left[{r}={2}{\sec{\theta}}\ {\quad\text{and}\quad}\ {r}={3}{\csc{\theta}}\right]}$$ are lines " is true or false.
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius $$\displaystyle{R}={7.4}\times{10}^{{-{15}}}$$ m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit $$\displaystyle={1.66}\times{10}^{{-{27}}}$$ kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).