Question

R is a vector space over C with respect to the usual operation of addition and multiplication. true or false.

Answer 1

Answer to Question #90985 – Math – Linear Algebra

Question

R is a vector space over C with respect to the usual operation of addition and multiplication. true or false.

Solution

R is not a vector space over C with respect to the usual operations.

Why? Recall that, if R would be a vector space over C, this means that any pair (r,c) is closed under scalar multiplication, where r is from R and c is from C.

This means that cr ∈ R for any complex c and any real vector r.

If we take, for instance, c = i and r=1 (read vector r) => cr = i and definitely i is not in R, since it is a complex number.

Therefore, it is not closed under multiplication => R is not a vector space.

It does not even worth checking the rest of the axioms.

Answer: R is not a vector space over C with respect to the usual operations.

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