Bilinear map

In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, is linear in each of its arguments. Matrix multiplication is an example. Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function B: V × W → X such that for all w ∈ W, the map B w v ↦ B is a linear map from V to X, for all v ∈ V, the map B v w ↦ B is a linear map from W to X. In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, for when we hold the second entry fixed; such a map B satisfies the following properties. For any λ ∈ F, B = B = λ B; the map B is additive in both components: if v 1, v 2 ∈ V and w 1, w 2 ∈ W B = B + B and B = B + B. If V = W and we have B = B for all v, w in V we say that B is symmetric. If X is the base field F the map is called a bilinear form, which are well-studied; the definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R.

It generalizes to n-ary functions. For non-commutative rings R and S, a left R-module M and a right S-module N, a bilinear map is a map B: M × N → T with T an -bimodule, for which any n in N, m ↦ B is an R-module homomorphism, for any m in M, n ↦ B is an S-module homomorphism; this satisfies B = r ⋅ B B = B ⋅ sfor all m in M, n in N, r in R and s in S, as well as B being additive in each argument. A first immediate consequence of the definition is that B = 0X whenever v = 0V or w = 0W; this may be seen by writing the zero vector 0V as 0 ⋅ 0V and moving the scalar 0 "outside", in front of B, by linearity. The set L of all bilinear maps is a linear subspace of the space of all maps from V × W into X. If V, W, X are finite-dimensional so is L. For X = F, i.e. bilinear forms, the dimension of this space is dim V × dim W. To see this, choose a basis for V and W. Now, if X is a space of higher dimension, we have dim L = dim V × dim W × dim X. Matrix multiplication is a bilinear map M × M → M. If a vector space V over the real numbers R carries an inner product the inner product is a bilinear map V × V → R.

In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × V → F. If V is a vector space with dual space V∗ the application operator, b = f is a bilinear map from V∗ × V to the base field. Let V and W be vector spaces over the same base field F. If f is a member of V∗ and g a member of W∗ b = fg defines a bilinear map V × W → F; the cross product in R3 is a bilinear map R3 × R3 → R3. Let B: V × W → X be a bilinear map, L: U → W be a linear map ↦ B is a bilinear map on V × U. Tensor product Sesquilinear form Bilinear filtering Multilinear map Multilinear subspace learning Hazewinkel, Michiel, ed. "Bilinear mapping", Encyclopedia of Mathematics, Springer Science+Business Media B. V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4


Großschirma is a town in the district of Mittelsachsen, in the Free State of Saxony, Germany. It is situated 7 km northwest of Freiberg, it was formed from the administrative union of the village of Großschirma and the town of Siebenlehn, including their districts, on 1 September 2003. The eponymous village Großschirma was founded in the middle of the 12th century. Claims by local historians that it was founded in 956 could not be proved; until the Protestant Reformation Großschirma belonged to Altzella Abbey. In 1555, prince-elector Augustus sold Großschirma and 14 other villages that were part of the estate of the dissolved abbey to his councillor Ulrich von Mordeisen. Rudolph Mordeisen, one of the sons of the latter, sold Großschirma and at least nine more villages in the property of his family to prince-elector Christian. Großschirma benefited from the growth of mining in the Freiberg Mining Field; the state-owned mine Churprinz Friedrich August Erbstolln on the west bank of the river Freiberger Mulde was one of the most profitable mines in the field.

The ore extracted there was conveyed to the smelters in Halsbrücke on an artificial waterway, the Churprinzer Bergwerkskanal. In 2003, the hitherto separate town Siebenlehn offered to join the municipality of Großschirma due to financial difficulties; the two municipalities merged on 1 September 2003, whereby the town privileges of Siebenlehn were transferred to the joint municipality. This was the first case of a town being incorporated into a rural municipality in Saxony. Bundesstraße 101 traverses the area of the municipality from north to south. Motorway A4 runs with a junction north-west of Siebenlehn. Nossen–Moldau railway traverses the town from north to south and has stations in Großvoigtsberg and Großschirma, it is principally used by occasional freight trains. A network of hiking paths and cycling routes, integrated into long-distance routes is maintained by local associations. Otto Rühle born in Siebenlehn. Formost radical Marxist Associated with Rosa Luxemburg and Alfredd Ardler. Amalie Dietrich, born in Siebenlehn, botanist and plant hunter Friedrich Wilhelm Putzger, born in Siebenlehn, schoolbook author

Arsenal de Belles Melodies

"Arsenal De Belles Melodies" is the second studio album by Congolese singer Fally Ipupa. It contains 16 tracks, it is the second album by Fally Ipupa to be produced by David Monsoh. On the album, Ipupa collaborated with Olivia Longott, a former member of 50 Cent's G-Unit musical group, on the song Chaise Électrique and with Krys on the song Sexy Dance; the album is the second album by Ipupa to go gold for selling over 100,000 copies in less than a month, including 40,000 sales in a week. This was a record for a Congolese artist. 2009Biçarbonate – 9:29 Çadenas – 8:24 Tsho – 6:28 Travelling Love – 8:23 Une minute – 7:24 Délibération – 7:03 Chaise Électrique – 3:55 Nyokalessé – 7:53 Mon amour – 6:15 Çatafalque – 8:03 La jungle – 5:46 Arsenal de belles mélodies – 9:40 5e race – 8:04 Orphelin amoureux – 5:09 Lourdes – 8:46 Sexy Dance – 3:21 "Chaise Électrique".