Weird field notation

Question

I have a question:

Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define

$$S = \{x\in\mathbb{F}^5 \mid x_i = x_{6-i}, 1 \leq i \leq 5 \}$$

$$T = \{x\in\mathbb{F}^5 \mid x_1 = x_4 = 0\}$$

Prove that $S$ is a subspace of $\mathbb{F}^5$

So like is this meaning like $5$-tuples? $(a,b,c,d,e)$? What is this $\mathbb{F}^5$ stuff?

A field characteristic = 0 just means a determinant is not invertible right?

Answer 1

Some help:

$$\mathbb{F}^5=\{(x_1,x_2,x_3,x_4,x_5)| x_i\in\mathbb{F}\}.$$

So, you need to show that $S=\{(a,b,c,b,a)|a,b,c\in\mathbb{F}\}$ is a subspace. Can you do that?

Answer 2

$\mathbb F^5$ is the $\mathbb F$-vector space of all $5$-tuples over $\mathbb F$.

Characteristic $0$ means that in $\mathbb F$, $$1 + 1 \neq 0,\quad 1 + 1 + 1\neq 0,\quad 1 + 1 + 1 + 1\neq 0,\ldots$$