Introduction to Higher Algebra

Introduction to Higher Algebra

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1907 edition. Excerpt: ...that we have a real quadratic form which vanishes at no real points except its vertices and the point (0, 0, ... 0). If it were indefinite, we could (Theorem 1) find two real points y, z, at one of which it is positive, at the other, negative. Hence (Theorem 2) we could find two real points linearly dependent on y and z, at which the quadratic form vanishes. Neither of these will be the point (0, 0, . 0), since, by Theorem 2, they are linearly independent. Moreover, they are neither of them vertices. Thus we see that the form must be definite, and the sufficiency of the condition is established. It remains to be proved that a definite form can vanish only at its vertices and at the point (0, 0, .. 0). Suppose (4) is definite and that (yv. yn) is any real point at which it vanishes. Then, n n n 2a, /z( + yi)(xj + yf) = Sa: c, + 2 Xlax. If y were neither a vertex nor the point (0, 0, ... 0), Saa; would not vanish identically, and we could find a real point (zv.. zn) such that n k = 2a _z#, 0. n If we let c = SdyZfy, we have (8) a0z, + X#)(-, + X%) = c + 2 7Jc. For a given real value of X, the left-hand side of this equation is simply the value of the quadratic form (4) at a certain real point. Accordingly, for different values of X it will not change sign, while the right-hand side of (8) has opposite signs for large positive and large negative values of X. Thus the assumption that y was neither a vertex nor the point (0, 0, 0) has led to a contradiction; and our theorem is proved. Corollary. A non-singular definite quadratic form vanishes, for real values of the variables, only when its variables are all zero. As a simple application of the last corollary we will prove Theorem 4. In a non-singular definite form, none of the coejfi dentsshow more

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  • Paperback | 90 pages
  • 189 x 246 x 5mm | 177g
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236563352
  • 9781236563354