By S. Fenyo, Thomas Frey

ISBN-10: 0720423686

ISBN-13: 9780720423686

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**Example text**

This contradiction proves the hypothesis. This means that if the mapping 31 has the property δ, 3ί simultaneously m generates a unique mapping o f Ρ onto the space R , for there is precisely one m vector χ in R that corresponds to each vector y in P. The mapping is linear : if yx and y2 are the image vectors of x1 and x2, then yx=Ax y2 = l 9 Ax2 whence yi +y2 + x 2) . = A(xl Thus the vector x x + x2 corresponds to the vector yx + y2. Moreover, if y is the image o f χ and λ is a scalar, then the equation y = A χ implies that Ày = A(Àx), or in other words that the vector λ χ corresponds to the vector Ay.

It is obvious that none of these vectors may be 0, as this would involve linear dependence. Linear independence requires that any arbitrary number of vectors in the system must themselves be linearly independent. 09: If we take suitable linear combinations of the given linearly independent vectors, we can form an orthogonal set of vectors. W e shall prove the theorem by giving a practical method of forming a vector system with the required properties. 06 29 ORTHOGONAL A N D BIORTHOGONAL SYSTEMS OF VECTORS W e shall in fact derive a second system o f vectors α ΐ 9α 2 , .

_! < K r - i . 069). It is also clear that the trapezoidal nature of U and V is unaffected by the construction o f linear combinations o f the type ocvf + 0*. 069) is the least one and thus that the vectors ut and t> f(/= 1, 2 , r ) are linearly independent. 06 that the determinant of an r-rowed minor o f U—for example, the one that consists of the first r rows—is non-zero. Likewise it follows (cf. 06a) that an r-rowed square minor, thus for example the one formed from the first r columns o f V, has a non-zero determinant.

### Modern Mathematical Methods in Technology: v. 2 by S. Fenyo, Thomas Frey

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