By Synowka-Bejenka E., Zontek S.

Within the paper the matter of simultaneous linear estimation of mounted and random results within the combined linear version is taken into account. an important and enough stipulations for a linear estimator of a linear functionality of fastened and random results in balanced nested and crossed category versions to be admissible are given.

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A group is a set G with a rule of multiplication; thus if g\ and gi are elements of G, there is a unique product gig2 in G. T h e following axioms must be satisfied: (a) (0102)03 = (0102)03, {associative law): (b) there is an identity element 1Q with the property : (c) each element g of G has an inverse element 0 _ 1 in G such t h a t gg~l = 1Q = 9'1g1G0 = 0 = 0 1 G These statements must hold for all elements g, gi, 02, 03 of G. Thus the set GLn (R) of all invertible matrices over R, a ring with identity, is a group; this important group is known as the general linear group of degree n over R.

L in the Appendix ]. 4. Let A = /l 0 \0 1 1\ 1 1 1 0/ and B = / O i l 1 1 1 \1 1 0 be matrices over the field of two elements. Compute A + B, A2 and AB. 5. Show that the set of all n x n scalar matrices over R with the usual matrix operations is a field. 6. Show that the set of all non-zero nxn scalar matrices over R is a group with respect to matrix multiplication. 7. Explain why the set of all non-zero integers with the usual multiplication is not a group. Chapter Two SYSTEMS OF L I N E A R EQUATIONS In this chapter we address what has already been described as one of the fundamental problems of linear algebra: to determine if a system of linear equations - or linear system - has a solution, and, if so, to find all its solutions.

9. Show that each elementary row operation has an inverse which is also an elementary row operation. 3 E l e m e n t a r y M a t r i c e s An nxn matrix is called elementary if it is obtained from the identity matrix In in one of three ways: (a) interchange rows i and j where i ^ j ; (b) insert a scalar c as the (i,j) entry where % ^ j ; (c) put a non-zero scalar c in the (i, i) position. 1 Write down all the possible types of elementary 2 x 2 matrices. These are the elementary matrices that arise from the matrix 12 = ( o i)'they are Ei=[ I l ) , E 2 = ( l 0 [),**=( I I and * - « s : ) • * - ( * °e Here c is a scalar which must be non-zero in the case of E4 and E5.