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**Extra resources for Axial symmetry of solutions to semilinear elliptic equations in unbounded domains**

**Sample text**

We also have the following. 2 A nontrivial vector space = over an infinite field - is not the union of a finite number of proper subspaces. Proof. Suppose that = ~ : r Ä r : , where we may assume that : \ : r Ä r : Let $ : ± ²: r Ä r : ³ and let # ¤ : . Consider the infinite set ( ~ ¸$ b # - ¹ which is the “line” through #, parallel to $. We want to show that each : contains at most one vector from the infinite set (, which is contrary to the fact that = ~ : r Ä r : . This will prove the theorem.

Then Ä , which implies that for some . We can assume by reindexing if necessary that ~ . Since is irreducible must be a unit. Replacing by and canceling gives Äc ~ Äc This process can be repeated until we run out of 's or 's. If we run out of 's first, then we have an equation of the form " Ä ~ where " is a unit, which is not possible since the 's are not units. Fields For the record, let us give the definition of a field (a concept that we have been using).

Hence, 1) implies 2). c Ä c ! 44 Advanced Linear Algebra Then 2) implies that ~ ~ and implies 3). " for all " ~ Á Ã Á . Hence, 2) Finally, suppose that 3) holds. If £ # : q then # ~ : q £ s t : and where p ~ bÄb : are nonzero. 5 Any matrix ( C can be written in the form (~ ²( b (! ³ b ²( c (! 1) where (! is the transpose of (. 1) is a decomposition of ( as the sum of a symmetric matrix and a skew-symmetric matrix. Since the sets Sym and SkewSym of all symmetric and skew-symmetric matrices in C are subspaces of C , we have C ~ Sym b SkewSym Furthermore, if : b ; ~ : Z b ; Z , where : and : Z are symmetric and ; and ; Z are skew-symmetric, then the matrix < ~ : c :Z ~ ; Z c ; is both symmetric and skew-symmetric.