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2. Complex number - Wikipedia

   en.wikipedia.org/wiki/Complex_number

   More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation (+) = has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions + and .

3. Linear algebra - Wikipedia

   en.wikipedia.org/wiki/Linear_algebra

   In multilinear algebra, one considers multivariable linear transformations, that is, mappings that are linear in each of a number of different variables. This line of inquiry naturally leads to the idea of the dual space , the vector space V* consisting of linear maps f : V → F where F is the field of scalars.

4. Algebra over a field - Wikipedia

   en.wikipedia.org/wiki/Algebra_over_a_field

   In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra. A left ideal of a K -algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace.

5. Linear complex structure - Wikipedia

   en.wikipedia.org/wiki/Linear_complex_structure

   The fundamental example of a linear complex structure is the structure on R 2n coming from the complex structure on C n.That is, the complex n-dimensional space C n is also a real 2n-dimensional space – using the same vector addition and real scalar multiplication – while multiplication by the complex number i is not only a complex linear transform of the space, thought of as a complex ...

6. Resolvent set - Wikipedia

   en.wikipedia.org/wiki/Resolvent_set

   In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism .

7. Scalar (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Scalar_(mathematics)

   A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.

8. Fredholm alternative - Wikipedia

   en.wikipedia.org/wiki/Fredholm_alternative

   The Fredholm alternative is the statement that, for every non-zero fixed complex number, either the first equation has a non-trivial solution, or the second equation has a solution for all (). A sufficient condition for this statement to be true is for K ( x , y ) {\displaystyle K(x,y)} to be square integrable on the rectangle [ a , b ] × [ a ...

9. Field (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Field_(mathematics)

   Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory.