- Arts & Culture 5905
- Books in Foreign Languages 214
- Business & Economics 4481
- Computers 2197
- Dictionaries & Encyclopedias 634
- Education & Science 96709
- Abstracts 1141
- Astrology 12
- Astronomy 13
- Biology 31
- Chemistry 5124
- Coursework 4070
- Culture 30
- Diplomas 2536
- Drawings 1480
- Ecology 30
- Economy 343
- English 1177
- Entomology 2
- Ethics, Aesthetics 22
- For Education Students 25977
- Foreign Languages 115
- Geography 19
- Geology 17
- History 315
- Maps & Atlases 36
- Mathematics 14530
- Musical Literature 7
- Pedagogics 234
- Philosophy 190
- Physics 15382
- Political Science 126
- Practical Work 133
- Psychology 468
- Religion 44
- Russian and culture of speech 102
- School Textbooks 69
- Sexology 68
- Sociology 48
- Summaries, Cribs 730
- Tests 20727
- Textbooks for Colleges and Universities 530
- Theses 190
- To Help Graduate Students 24
- To Help the Entrant 114
- Vetting 430
- Works 59
- Информатика 14

- Engineering 3302
- Esoteric 952
- Fiction 3285
- For Children 418
- House, Family & Entertainment 2501
- Law 2826
- Medicine 1148
- Newspapers & Magazines 344
- Security 281
- Sport, Tourism 1001
- Website Promotion 653

# MIREA. Typical calculation-2 on Linear Algebra. Var-17

Uploaded:

**20.02.2014**

Content: mirea_linalg_2_v17.rar 589,37 kB

## Product description

E-book (DjVu-file) contains solutions of 8 typical problems for the first-year students of full-time education. Problems are taken from the from the task book in Algebra and Geometry developed for MIREA students. Authors: I.V.Artamkin, S.V.Kostin, L.P.Romaskevich, A.I.Sazonov, A.L.Shelepin. Yu.I.Hudak Editor (Publisher MIREA 2010). Variant-17.

Problem solutions are presented in the form of scanned handwriting papers collected into a single document of 21 pages. This document is saved in the DjVu-format which can be opened in the Internet Explorer or Mozilla Firefox browsers with the aid of the DjVu plug-in. Links to download and to install DjVu plug-in are attached. DjVu-file containing the problems and their detailed solutions is ready for viewing on a computer and for printing. All solutions were successfully accepted by MIREA teachers.

Problems of the Typical calculation:

Problem 1. Find fundamental system of solutions and the general solution of the homogeneous system of equations.

Problem 2. Find the general solution depending on the value of parameter λ. For what values of λ system admits a solution with the aid of the inverse matrix?

Problem 3. A linear operator A: V3 - V3 is determined by its action on the ends of the radius vectors of three-dimensional space of points.

1) Find the matrix of the operator A in a suitable basis V3 space, and then in the canonical basis.

2) Determine to what points are moving the point with coordinates (1,0,0) and (-1,2,1) under the action of α.

Problem 4. Let A - matrix of A of task 3 in the canonical basis. Find the eigenvalues and eigenvectors of A. Explain how the result is related to the geometrical effect of A.

Problem 5.

1) Prove that A is a linear operator in the space Pn of polynomials of degree n.

2) Find the matrix of the operator A in the canonical basis Pn.

3) Do the inverse operator A-1 exist? If so, find his matrix.

4) Find the image, the core of the rank and defect of A.

Problem 6. The operator A acts on the matrix, forming a linear subspace M in the space of matrices of order.

1) Prove that A - line operator in M.

2) Find the matrix of A in some basis M.

3) Find the image, the core of the rank and defect of A.

Problem 7. In the space of geometrical vectors V3 with the usual scalar product of basis vectors are set coordinates in the canonical basis.

1) Find the matrix of Gram GS scalar product in this basis. Write out the formula for the length of the vector through its coordinates in the basis S.

2) Orthogonalize the basis S. Make checks of orthonormality for the built basis P in two ways:

a) by writing out the coordinates of the vectors in the basis of P;

b) by ensuring that the conversion of the Gram matrix of the transition from the base S to the base P (use Formula GP = CT*GS*C) leads to an identity matrix.

Problem 8. Given a quadratic form .

1) Bring it to the canonical form with the Lagrange method. Record the corresponding transformation of variables.

2) Bring it to the canonical form by using an orthogonal transformation, write the transition matrix.

3) Verify the validity of the law of inertia of quadratic forms in the transformation example, received in paragraphs 1 and 2.

4) The surface of the second order σ is set in a rectangular Cartesian coordinate system by the equation Q(x)=α. Determine the type of surface σ, and write its canonical equation.

## Additional information

The document was prepared on the web-site:

Web-Tutor in Physics and Mathematics.

Problems and other information may be found on the Web-Tutor site in section

MATHEMATICS.