MIREA. Typical calculation-2 on Linear Algebra. Var-19

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-19.

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.