Vector Spaces

6.1

Suppose that

$$S = \{ax^2 + bx + c : a, b, c \in \mathbb{R}, a > 0\}$$

is the set of quadratic functions which have graphs opening upwards.
Define addition of functions by the usual rule, so that for example

$$(3x^2 - 2x + 1) + (5x^2 + 4x + 17) = 8x^2 + 2x + 18,$$

and define scalar multiplication also by the usual rule, so that for example

$$5(3x^2 - 2x + 1) = 15x^2 - 10x + 5.$$

Which of the following 10 properties of a vector space are satisfied?

1. Since $a>0$, any $a_1+a_2 > 0$, $b,c$ do not matter hence this is satisfied.
2. Self-explanatory, satisfied
3. Self-explanatory, satisfied
4. Since $a>0$, there can not be a $\mathbf{0}$ vector, not satisfied
5. Since $a>0$, there cannot be a negative vector, not satisfied
6. Suppose $\lambda = -1$, $\lambda(ax^2+bx+c)=-ax^2-bx-c$, $a<0$, meaning it is no longer in $S$, not satisfied
7. Self-explanatory, satisfied
8. Self-explanatory, satisfied
9. Self-explanatory, satisfied
10. Self-explanatory, satisfied

The 13th century Italian mathematician Leonardo of Pisa was also known as Fibonacci.
Sometimes we like to consider sequences $\mathbf{a} = (a_1, a_2, a_3, \cdots)$, where $a_i \in \mathbb{Q}$, just like vectors. Now consider the set of Fibonacci-type sequences, that is

$$F = \{(a_1, a_2, a_3, \cdots) : \text{where} \quad a_{n+2} = a_{n+1} + a_n \quad \text{for} \quad n \ge 1\}.$$

Define vector/sequence addition and scalar multiplication (over $\mathbb{Q}$) of elements of $F$ component-wise, so that for example

$$(1, 1, 2, 3, 5, 8, \cdots) + (2, 4, 6, 10, 16, 26, \cdots) = (3, 5, 8, 13, 21, 34, \cdots)$$

and

$$3(1, 1, 2, 3, 5, 8, \cdots) = (3, 3, 6, 9, 15, 24, \cdots).$$

Which of the following 10 properties of a vector space are satisfied?
We only have to check $1,4,5,6$ due to the subspace theorem,

1. If you have two valid Fibonacci sequences $\mathbf{a}$ and $\mathbf{b}$, and you add them together to make a new sequence $\mathbf{c}$, does it still follow the rule? Yes. $c_{n+2} = a_{n+2} + b_{n+2} = (a_{n+1} + a_n) + (b_{n+1} + b_n)$.
2. The 0 vector $(0,0,0, \cdots)$ is valid since $0+0=0$.
3. If you multiply a Fibonacci sequence by a scalar $k$, every term is multiplied by $k$. Does $ka_{n+2} = ka_{n+1} + ka_n$? Yes, you can just factor the $k$ out of the right side to see it perfectly holds.
4. This is just property 6 using the specific scalar $k = -1$. Because scalar multiplication works, multiplying a sequence by $-1$ gives a valid sequence that perfectly cancels out the original.

6.3