6—Vector Spaces 136Exercises1 Determine if these are vector spaces with the usual rules for addition and multiplication by scalars.
If not, which axiom(s) do they violate?
(a) Quadratic polynomials of the formax^2 +bx
(b) Quadratic polynomials of the formax^2 +bx+ 1
(c) Quadratic polynomialsax^2 +bx+cwitha+b+c= 0
(d) Quadratic polynomialsax^2 +bx+cwitha+b+c= 1
2 What is the dimension of the vector space of (up to) 5th degree polynomials having a double root
atx= 1?
3 Starting from three dimensional vectors (the common directed line segments) and a single fixed
vectorB~, is the set of all vectors~vwith~v.B~= 0a vector space? If so, what is it’s dimension?
Is the set of all vectors~vwith~v×B~= 0a vector space? If so, what is it’s dimension?
4 The set of all odd polynomials with the expected rules for addition and multiplication by scalars. Is
it a vector space?
5 The set of all polynomials where the function “addition” is defined to bef 3 =f 2 +f 1 if the number
f 3 (x) =f 1 (−x) +f 2 (−x). Is it a vector space?
6 Same as the preceding, but for (a) even polynomials, (b) odd polynomials
7 The set of directed line segments in the plane with the new rule for addition: add the vectors
according to the usual rule then rotate the result by 10◦counterclockwise. Which vector space axioms
are obeyed and which not?