By Monge G.

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En /, which is an n2 -dimensional vector space. Proof. En /. 32), any linear combination aij ei ˝ ej D 0 implies that aij D 0 for all the indices i; j D 1; : : : ; n. En /. 1. 6. 1; 1/-tensor is a bilinear map T W En En ! En / of all mixed 2-tensors becomes a vector space by the introduction of the standard operations of addition of two mixed 2-tensors and the product of a mixed 2-tensor by a real number. 7. The tensor product of a vector x and a covector mixed 2-tensor x ˝ !. ; y/ D x. 4. Â i / be the dual basis in En .

X1 ; x1 / ¤ 0. u1 ; u1 / D ˙1, where the sign is chosen according to the sign of ˛. u1 ; v/ D 0g: It is En D U 1 ˚ V 1 . u1 ; u1 / so that v 2 V 1 . 5) is unique. In fact, if there is another decomposition x D v0 C bu1 , v0 2 V 1 , then we have v0 v 2 V 1 since V 1 is a vector subspace of En . a b/u1 and v0 v 2 U 1 \ V 1 . a b/˛: Since ˛ ¤ 0, the preceding condition implies a D b and v0 D v. Now we consider the restriction T1 of T over V 1 , which is a bilinear symmetric map. V 1 / D n 1 D 1, then the theorem is proved.

28 2 Tensor Algebra 9. 0; 2/-tensor T satisfy either of the conditions Tij D Tj i ; Tij D Tj i in a given basis, then they satisfy the same conditions in any other basis. 10. u; v/ 8v. 1. 1) 8x; y 2 En . x; x/ D 0. ei / of En . 1. 2) hold in a basis, then they hold in any basis. 2. : A. 1. 4) has the following properties: • ! , a 2 <, then ! ^ D 0. Proof. 4), 8x; y 2 En , we have that ! x; y/ D ! y/ D and the skew symmetry of ! ^ the other two properties. y/ ! 5) is proved. En /. En / are determined by the following theorem.