\displaystyle{14}{x}^{{2}}+{70}{x} Explanation: \displaystyle{\left({2}{x}+{10}\right)}\times{7}{x} means you multiply everything in the bracket by the term outside \displaystyle{\left({2}{x}+{10}\right)}\times{7}{x}={2}{x}\times{7}{x}+{10}\times{7}{x} ...

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First figure out what the lexicographic order topology on the plane is! What does an open interval look like? This should give you a basis to answer the subspace question. Added: you saw that the open ...

Yes. This is follows from the rank-nullity theorem . If Ax=b has no solution, it means that there is a vector that does not belong to the range of A. Hence, the dimension of the range of A ...

Hints \rm\ \ (1)\quad 3\cdot 33\,\mid\, 3\cdot 13x-65\ \Rightarrow\ 3\mid 65 \rm(2)\quad 20\mid 8x-12\iff 5\mid 2x-3\iff 5\mid 2x+2 = 2(x+1) \rm(3)\quad mod\ 43\!:\ \dfrac{1}{21}\equiv \dfrac{2}{42}\equiv \dfrac{2}{-1}\equiv -2 ...

Yes, that's right. The tensor product over k is the coproduct in the category of commutative k-algebras, and the polynomial ring in n variables over k is the free commutative k-algebra on n ...