The set $\color{gold}ℝ^𝚗$ is known as an Euclidean n-space

We may think of its elements $a = (a_1,a_2 ..,a_n)$ as vectors or n-vectors.

Standard basis vectors

\begin{matrix}
𝒖_1 = (1, 0, 0 & ⋯ & 0) \\
𝒖_2 = (0, 1, 0 & ⋯ & 0) \\
⋮ \\
𝒖_n = (0, 0, 0 & ⋯ & 1) \\

\end{matrix}

Subspaces

Let $W$ be a subset of the space vector $V$ . We say that $W$ is a subspace of $V$ if it satisfies the following properties:

• $W$ is different from the empty set

• $W$ is closed with respect to the sum, that is, for every $𝒖, 𝒗 ∈ W$ we have that $𝒖 + 𝒗 ∈ W$

• $W$ is closed with respect to the product by scalars, that is, for every $𝒖 ∈ W$ and every $λ ∈ ℝ$ we have that $λ𝒖 ∈ W$.

• We see that: $0_V ∈ W$

In particular, therefore, every vector space $V$ has always at least two subspaces: $V$ itself and the zero subspace, consisting of only the zero vector $0_V$ .

Example: ℝ (the set of reals) is a subset of ℂ (the set of complex numbers).

Example: $W$ = { $(x,y) ∈ ℝ^2 ∣ y = ax$ } subspace of $ℝ^2$

• $(0,0) ∈ W$
• $(x_1, a x_1) + (x_2, a x_2) = (x_1+x_2, a(x_1+x_2)) ∈ W$
• $𝜆(x_0, ax_0) = (𝜆x_0, a𝜆x_0)$

Example: In the Vector Space $ℝ[x]$ of polynomials with real coefficients in a variable $x$, consider the subset $ℝ_2[x]$ consisting of polynomials of degree less than or equal to $2$:
$ℝ_2[x]=$ { $p(x)=a+bx+cx^2 ∣ a,b,c ∈ ℝ$ }.

• $a+bx + cx^2 + a'+b'x + c' x^2 = (a+a')+(b+b')x+(c+c')x^2$
• $𝜆(a+bx + cx^2) = 𝜆a+ 𝜆b·x + 𝜆c·x^2$

The intersection $S_1 ∩ S_2$ of two subspaces, $S_1$ and $S_2$ of a vector space $V$, is also a subspace of $V$ .