This section establishes a connection between vector spaces and show that all finite-dimensional spaces are equivalent to 𝔽n, where 𝔽 is either ℚ (rational numbers) or ℝ (real numbers) or ℂ (complex numbers). This aloows us to extend vector operations from 𝔽n to a vector space. Therefore, any finite dimensional vector space has the same algebraic structure as 𝔽n even though its vectors may not be expressed as n-tuples.

Isomorphism

A linear transformation T : V ⇾ W between two vector spaces that is both one-to-one and onto is said to be an isomorphism, and W is said to be isomorphic to V, which is abbreviated as V ≌ W.

Observe that if T : X ⇾ Y is an isomorphism, then the inverse T−1 : Y ⇾ X is a linear map, hence also an isomorphism.
Theorem 1: Two finite-dimensional vector spaces are isomorphic precisely when they have the same dimension.
Let T : X ⇾ Y be an isomorphism. Since T is injective, kerT = {0}, and dim ker(T) = 0. Also since T is surjective, Im(T) = Y, so dim Im(T) = dim(Y). Hence \[ \dim\,X = dim\mbox{Im}T + \dim\,\mbox{ker}(T) = \dim\,Y. \] Conversely, if X and Y have the same dimension n, take bases {ei} of X and {εj} of Y. Since they have the same number of elements by assumption, we may index them by the same index set 1 ≤ i ≤ n. The mapping \[ {\bf x} = \sum_i x_i {\bf e}_i \ \mapsto \ \sum_i x_i \varepsilon_i \] defines a n isomorphism between X and Y.
Example 1: Every polynomial is uniquely identified by its coefficients. Therefore, we get the isomorphism \[ \mathbb{R}_{\le 2} [x] \ni a + b\, x + c\, x^2 \mapsto \left[ a, b, c \right] \in \mathbb{R}^3 . \]
End of Example 1
■
Theorem 2: Every finite-dimensional vector space of dimension n ≥ 1 is isomorphic to 𝔽n.
Let us choose a basis (ei), 1 ≤ i ≤ n, of the vector space X. Hence, each element x ∈ X has a unique representation as a linear combination of the ei's, say, \( \displaystyle {\bf x} = \sum_i x_i {\bf e}_i . \) Let f(x) denote the n-tuple formed by the the components of x: \[ f({\bf x}) = \left( x_1 , x_2 , \ldots , x_n \right) \in \mathbb{R}^n . \] This map is bijective by definition. It is linear and its inverse 𝔽n ⇾ X is \[ \left( x_i \right)_{1 \leqslant i \leqslant n} \mapsto \sum_{1 \leqslant i \leqslant n} x_i {\bf e}_i . \]
Example 5: The infinite set of monomials V₁ has the basis β = {[1, 0, 0], [0, 1, −1]} because every vector from V₁ is uniquely expanded as \[ \left[ a, b, -b \right] = a \left[ 1, 0, 0 \right] + b \left[ 0, 1, -1 \right] . \] This space is isomorphic to V₂ according to linear transformation \[ \phi \,V_1 \,\mapsto \, V_2 , \qquad \phi ([a, b, -b]) = \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \] that preserves vector operations (vector addition and scalar multiplication). Indeed, \[ \phi \left( \left[ a_1 + a_2 , b_1 + _2 , - b_1 - b_2 \right] \right) = \begin{bmatrix} a_1 + a_2 & b_1 b_2 \\ -b_1 - b_2 & a_1 + a_2 \end{bmatrix} = \begin{bmatrix} a_1 & b_1 \\ -b_1 & a_1 \end{bmatrix} + \begin{bmatrix} a_2 & b_2 \\ -b_2 & a_2 \end{bmatrix} = \phi ([ a_1 , b_1 , -b_1 ]) + \phi (a_2 , b_2 , -b_2 ]). \] This mapping also preserves scalar multiplication: \[ \phi \left( [\lambda a, \lambda b, -\lambda b]\right) = \begin{bmatrix} \lambda a & \lambda b \\ -\lambda b & a \end{bmatrix} = \lambda \begin{bmatrix} a & b \\ -b & a \end{bmatrix} = \lambda\,\phi \left( [a, b, -b] \right) . \] Any vector from V₂ can be uniquely expanded through two basis elements: \[ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} = a \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + b \begin{bmatrix} \phantom{-}0 & 1 \\ -1 & 0 \end{bmatrix} . \] We can also establish a linear transformation ψ V₂ ⇾ V₃ by formula \[ \psi \, : \,V_2 \mapsto V_2 , \qquad \psi \left( \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \right) = a + b\, x - b\,x^2 + a\,x^3 . \]
End of Example 5
■
The matrix description of a linear map T : X ⇾ Y between finite-dimensional vector spaces, is a consequence of choices of bases in both domain and target spaces. These choices are reflected by vertical isomorphisms