y=2x−3

dizxindlert7
2022-09-12
Answered

y=2x−3

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Find the linear approximation of the function

Use L(x) to approximate the numbers

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Find the slope that is perpendicular to 2x+8y=9

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Here is the example I encountered :

A matrix$M(5\times 5)$ is given and its minimal polynomial is determined to be $(x-2)}^{3$ . So considering the two possible sets of elementary divisors

$\{{(x-2)}^{3},{(x-2)}^{2}\}\text{}\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}\text{}\{{(x-2)}^{3},(x-2),(x-2)\}$

we get two possible Jordan Canonical forms of the matrix , namely$J}_{1$ and $J}_{2$ respectively. So $J}_{1$ has 2 and $J}_{2$ has 3 Jordan Blocks.

Now we are to determine the exact one from these two. From the original matrix M, we determined the Eigen vectors and 2 eigen vectors were linearly independent. So the result is that$J}_{1$ is the one .So, to determine the exact one out of all possibilities , we needed two information -

1) the minimal polynomial ,together with 2) the number of linearly independent eigen vectors.

Now this was a question-answer book so not much theoretical explanations are given . From the given result , I assume the number of linearly independent eigen vectors -which is 2 in this case - decided$J}_{1$ to be the exact one because it has 2 Jordan Blocks. So the equation

"Number of linearly independent eigen vectors=Number of Jordan Blocks"

must be true for this selection to be correct .

Now this equation is not proved in this book or the text book I have read says nothing of this sort

So, that is my question here : How to prove the equation "Number of linearly independent eigen vectors=Number of Jordan Blocks"?

A matrix

we get two possible Jordan Canonical forms of the matrix , namely

Now we are to determine the exact one from these two. From the original matrix M, we determined the Eigen vectors and 2 eigen vectors were linearly independent. So the result is that

1) the minimal polynomial ,together with 2) the number of linearly independent eigen vectors.

Now this was a question-answer book so not much theoretical explanations are given . From the given result , I assume the number of linearly independent eigen vectors -which is 2 in this case - decided

"Number of linearly independent eigen vectors=Number of Jordan Blocks"

must be true for this selection to be correct .

Now this equation is not proved in this book or the text book I have read says nothing of this sort

So, that is my question here : How to prove the equation "Number of linearly independent eigen vectors=Number of Jordan Blocks"?

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How many terms will the simplified ansequation wer have?

asked 2022-05-16

Suppose we have the following non-linear differential equation

$\ddot{x}+{\omega}^{2}(t)x-\frac{1}{{x}^{3}}=0$

with $x(t)$ being a real function (and $\omega (t)$ being also time-dependent).

Is there an analytical solution?

If not, is there an analytical solution for some particular form of $\omega (t)$?

If the answer is again no, what software would be recommendable for numerical solution?

$\ddot{x}+{\omega}^{2}(t)x-\frac{1}{{x}^{3}}=0$

with $x(t)$ being a real function (and $\omega (t)$ being also time-dependent).

Is there an analytical solution?

If not, is there an analytical solution for some particular form of $\omega (t)$?

If the answer is again no, what software would be recommendable for numerical solution?

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Determine if (1,3) is a solution to the given system of linear equations.

$5x+y=8$

$x+2y=5$

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How do you find the slope and intercept of 2x+5y=15?