Mat102

This submission must be made by hand, without any digital aids except the end of part 2a) and delivered as a pdf (for the last part of 2a you can cut the screenshot from the program and the result of the run and paste in the pdf file). This usually means that you write by hand and then scan your papers, or take pictures and use a tool to turn pictures into pdf (here is a website for how it can be done I found on google: https://gadgets.ndtv.com/how-to/features/jpg-to-pdf-converter-free-online-merge-i-love-pdf-app-download-joiner-2245453.)). You can of course also write the submission in word or using latex and export it as a pdf if you prefer.

Task 1

An inhomogeneous linear system of equations is given by:

Write the equation system on the form: Ax=b ,there x= . Use either

Gaussian elimination or the determinant of A to determine for which values of K and t have the system:

i) an unambiguous solution ii) infinite number of solutions iii) no solution.

b) An inhomogeneous linear system of equations is given by:

Write the equation system on the form: Ax=b ,there x= . Use Gaussian elimination to solve the equation system.

Task 2

Vi skal i denne oppgaven studere matrisen

Show that one of the values is -2 and find the corresponding vector. Find all the values of A and the associated vectors without computer tools and the procedure must be displayed. Find all the values of A and the associated vectors with a write a Python program.

We have not learned together the concept of diagonalization (A = PAP^(-1) ). Read notes from NTNU and write briefly what diagonalisation of square matrices is about, and what the matrices P and D are. Diagonalize the matrix A.

Do you find any connection between the answers you received in sub-tasks a) and b)?

Task 3

Newton's cooling law states that changing the temperature of an object is proportional to the difference between the temperature of the object and the environment. In a closed system that only consists of two objects, for example in two rooms with a door in between and no heat loss anywhere else, they are each other's surroundings. If we call the temperature of one object x(t) and the temperature of the other object y(t), then we have a model for the temperature of the two objects given as

Where m1 and m2 are positive constants.

Let m1=0.1 and m2=0.2 . Find the general solution for x(t) and y(t).

Regardless of which value we choose for m1 and m2, one value will be 0. What is the associated vector, and what is the physical explanation for this situation?

This submission must be made by hand, without any digital aids except the end of part 2a) and delivered as a pdf (for the last part of 2a you can cut the screenshot from the program and the result of the run and paste in the pdf file). This usually means that you write by hand and then scan your papers, or take pictures and use a tool to turn pictures into pdf (here is a website for how it can be done I found on google: https://gadgets.ndtv.com/how-to/features/jpg-to-pdf-converter-free-online-merge-i-love-pdf-app-download-joiner-2245453.)). You can of course also write the submission in word or using latex and export it as a pdf if you prefer.

Task 1

An inhomogeneous linear system of equations is given by:

Write the equation system on the form: Ax=b ,there x= . Use either

Gaussian elimination or the determinant of A to determine for which values of K and t have the system:

i) an unambiguous solution ii) infinite number of solutions iii) no solution.

b) An inhomogeneous linear system of equations is given by:

Write the equation system on the form: Ax=b ,there x= . Use Gaussian elimination to solve the equation system.

Task 2

Vi skal i denne oppgaven studere matrisen

Show that one of the values is -2 and find the corresponding vector. Find all the values of A and the associated vectors without computer tools and the procedure must be displayed. Find all the values of A and the associated vectors with a write a Python program.

We have not learned together the concept of diagonalization (A = PAP^(-1) ). Read notes from NTNU and write briefly what diagonalisation of square matrices is about, and what the matrices P and D are. Diagonalize the matrix A.

Do you find any connection between the answers you received in sub-tasks a) and b)?

Task 3

Newton's cooling law states that changing the temperature of an object is proportional to the difference between the temperature of the object and the environment. In a closed system that only consists of two objects, for example in two rooms with a door in between and no heat loss anywhere else, they are each other's surroundings. If we call the temperature of one object x(t) and the temperature of the other object y(t), then we have a model for the temperature of the two objects given as

Where m1 and m2 are positive constants.

Let m1=0.1 and m2=0.2 . Find the general solution for x(t) and y(t).

Regardless of which value we choose for m1 and m2, one value will be 0. What is the associated vector, and what is the physical explanation for this situation?

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