By Niçaise

ISBN-10: 2100049410

ISBN-13: 9782100049417

Nicaise S. examine numerique et equations aux derivees partielles (Dunod, 2000)(fr)(ISBN 2100049410)

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**Example text**

X,, has for a solution a set of s quantities Xi, ••• , x, which, on substitution in (1), reduce the left sides to k; (i = 1, ... , n). Thus a solution is a vector belonging to the a-dimensional space V,(iJ). This idea fits in well with the matrix notation for the system. If C = (c;;), X = col (xi, ... •. ), the system becomes CX=K. In this form the system may be regarded as having only one unknown, the vector X. Two systems of n linear equations in s unknowns Xi, . . , x. are called equivalent if every solution of one system is also a solution of the other, and vice versa.

Moreover, by property (c) of Theorem 3-3, 1 s c1 < c. < · · · < c, s r. 48 EQUIVALENCE, RANK, AND INVERSES [CHAP, 3 It follows that = 1, Ct = 2, ... 1 C,. = T. The ith equation begins with the term x,, = x,, and contains no term x,;,j ¢ i, by property (d) of Theorem 3-3. Thus equation i contains only the term x, and must be the equation Xi= 0. (i = 1, ... , r) The assumption that r = s thus leads to the conclusion that only the trivial solution is possible. This covers all the possibilities and completes the proof.

A basis a1, ... , ar of V. 2--6B. If V has dimenlfion r, any set of r + 1 vectors belonging to V is dependent. COROLLARY The proof is parallel to that of Theorem 2-5. 2--6C. Let V have dimenlfion r. Then r vectors in V form a basis of V if and only if they are linearly independent. COROLLARY If the r vectors are linearly independent but are not a basis, Corollary 2--6A provides a basis with more than r vectors, contrary to Theorem 2--6. Thus the r vectors must form a basis. The converse is trivial.

### Analyse numérique et équations aux dérivées partielles: cours et problèmes résolus by Niçaise

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