By D. E. Littlewood

ISBN-10: 0486627152

ISBN-13: 9780486627151

**Read or Download A University Algebra: An Introduction to Classic and Modern Algebra PDF**

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**Additional info for A University Algebra: An Introduction to Classic and Modern Algebra**

**Sample text**

Only if this is zero is the rank less than the number o f rows. The next step might conveniently be to find the number o f linearly independent rows by considering linear combina tions which have zero in one particular position. As an example find the rank of the matrix "1 , 1, 1, 1 ~ 1, 3, — 2, a 2, 2a - 2, - o- 2, 3a — 1 _3, a-f-2, — 3, 2a -J- 1 _ for all values of a. 1 a 3a - 1 2a + 1 0, i, 2, i, 2, 3, 0 - 1 , E? i 1, l, 1, 3, - 2, l, 2, 2a - 2, - a — 2, 3, a + 2, — 3, 1 The determinant is o, - 3, a - 4, - 6, 0 a - 1 3a — 3 2a - 2 33 MATRICES = (O - - 3, 1 0, o, 1 2, 1) 2a — 4, — a — 4, 3 = (o - 1) 2a — 10, — a + 5, 3 a — 5, 0 -1 , - 6, 2 0, 2 = (a - l)(a — 6)2.

Then the coefficient o f x is o^Ai a2A 2 -}- dzA z — A , the coefficient o f y is b-±Ai — b2A 2 -J- bzA z = 0, and o f z, c2A 2 “h ^3^3 = 0, while the coefficient o f w is d1A 1 — d%A2 -j- dzA z — b\9 cl9 d1 b29 ^2> d2 • bZ9 c3, dz The equation thus gives the ratio o f a; to w in accordance with the theorem. The ratios y : w and z : w are obtained similarly b y multi plying the original equations respectively by B l9 — B 2i B z and by Cl9 — C2i Oz. The cases so far considered have concerned only homogeneous linear equations.

The first row of the matrix is linearly independent of them all. Hence if a = 1, the rank is 3. For a = 5, the matrix is ri, l, 1, 1" 1, 3 , - 2 , 5 2, 8, - 7, 14 * _3, 7, - 3, 11_ Thence row (2) — row (3) — row (4) — row (1) = [0, 2, — 3, 4], 2 row (1) = [0, 6, — 9, 12], 3 row (1) = [0, 4, — 6, 8]. The rank of this set is clearly 1, hence the rank of the original matrix when a = 5 is 2. Consider the general case of n linear equations in m unknowns, <*2 1* 1 + <*1 2 * 2 <*im*m = *j" <*2 2 * 2 + • • • • + <*2 mXm ~ bi> b2> “f" ®«2 * 2 + ------ ”J” <*«m*m = <*»• It is necessary for consistency that the ranks of the matrices an, <*l 1, • • * * * <*1m — <*nl» • • • •> <*nm— * * • *> <*i7n> b l _ a 7 n l > ------- > a n m > b n — should be equal.

### A University Algebra: An Introduction to Classic and Modern Algebra by D. E. Littlewood

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