Mathematics in Ancient and Medieval India by A.K.Bag

By A.K.Bag

Heritage of arithmetic in historical and medieval India

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Mathematics in Ancient and Medieval India

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Xn ) ∈ Rn and f (x) = x1 + ... + xn then Hf = {x ∈ Rn : x1 + ... + xn = 1}. Determine the general form of hyperplanes in Rn containing zero. 5. Give examples of linear functionals in the spaces Fn [t], F[t], C[a, b] and determine the corresponding hyperplanes (cf. 9). 6. Mappings preserving all linear manifolds are called affine transformations. Describe all affine transformations in the spaces R2 and R3 . 7. ,p . ,n Prove that (A + B)T = AT + B T , (tA)T = tAT f or t ∈ F, (AT )T = A, (CA)T = AT C T (cf.

4, det A = det B = − det A which implies 2 det A = 0 and det A = 0. A similar proof holds for rows. 6. A determinant does not change its value if to elements of its column (row) there are added elements of another column (row) multiplied by an arbitrary number c. Proof. anpn = det A + c det A . ,pn } But det A has two identical columns: the jth column and the kth column. 5 together imply that det A = 0. A similar proof holds for rows. A minor determinant of a square matrix A of dimension n is said to be a determinant of A obtained by canceling the same number of columns and rows in A.

The system changes its sign. Systems of linear equations 37 Indeed. , an } = = sign (aj − ak ) sign (ar − as ) sign (aj − ar )sign (ar − ak ) . ,n r=j,k Observe that the interchange of the term aj and the term ak does not change the sign of the last two products. 3). , n. ,n is denoted also in the following way: a11 a det A = 12 ... a1n a21 a22 ... a2n ... an1 ... an2 . ... ann Columns and rows of a determinant det A are, by definition, columns and rows of the matrix A. 1. If n = 1 then det A = |a11 |.

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