- Arts & Culture 4968
- Books in Foreign Languages 205
- Business & Economics 4687
- Computers 2361
- Dictionaries & Encyclopedias 670
- Education & Science 82132
- Abstracts 1263
- Astrology 13
- Astronomy 13
- Biology 39
- Chemistry 3835
- Coursework 3731
- Culture 32
- Diplomas 2595
- Drawings 1697
- Ecology 31
- Economy 329
- English 1253
- Entomology 2
- Ethics, Aesthetics 29
- For Education Students 23167
- Foreign Languages 122
- Geography 20
- Geology 17
- History 231
- Maps & Atlases 41
- Mathematics 6324
- Musical Literature 5
- Pedagogics 229
- Philosophy 190
- Physics 12931
- Political Science 132
- Practical Work 111
- Psychology 492
- Religion 50
- Russian and culture of speech 103
- School Textbooks 69
- Sexology 67
- Sociology 53
- Summaries, Cribs 763
- Tests 20860
- Textbooks for Colleges and Universities 546
- Theses 188
- To Help Graduate Students 24
- To Help the Entrant 112
- Vetting 356
- Works 58
- Информатика 9

- Engineering 3241
- Esoteric 1136
- Fiction 3197
- For Children 422
- House, Family & Entertainment 2641
- Law 2828
- Medicine 1239
- Newspapers & Magazines 336
- Security 315
- Sport, Tourism 986
- Website Promotion 694

# MIREA. Typical calculation-2 on Linear Algebra. Var-14

Refunds: 0

Uploaded:

**20.02.2014**

Content: mirea_linalg_2_v14.rar (561,82 kB)

# 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-14.

Problem solutions are presented in the form of scanned handwriting papers collected into a single document of 19 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.