Question

# Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For thos

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Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The function are real-valued. All rational functions.

2021-07-03

Axiom 1: Clearly the sum of two rational functions Is again a rational function.
Axiom 2: If p(x) is a polynomial, then so is op(x) where @ is a real number.
Hence $$\displaystyle{a}{\left({}\frac{{{p(x)}}}{{{q}{\left({x}\right)}}}\right)}$$ is a rational function for all real a.
Axiom 3: Becatise addition of functions is defined pointwise, this follows from commutativity of addition of real numbers.
Axiom 4: Same for associativity.
Axiom 5: Notice that the zero function $$(x) = 0$$ is « rational function because it is a polynomial whose every coefficient is 0.
Axiom 6: Follows from existence of negatives in the reals.
Axiom 7: Follows from associativity of multiplication of the reals.
Axiom 8: Follows from distributive law of the reals.
Axiom 9: Also follows from the distributive law of the reals.
Axiom 10: $$\displaystyle{1}{\left({}\frac{{{p(x)}}}{{{q}{\left({x}\right)}}}\right)}=\frac{{{p}{\left({x}\right)}}}{{{q}{\left({x}\right)}}}$$ for all polynomials p and q.