Isomorphism

Definition: Two vector spaces $V$ and $W$ over the same field $F$ are isomorphic (written $V \cong W$) if there exists a bijective linear transformation $T: V \to W$.

The Fundamental Isomorphism Theorem for Vector Spaces: Every finite-dimensional vector space $V$ of dimension $n$ over $F$ is isomorphic to $F^n$.

Proof sketch: Choose a basis $\{v_1, \ldots, v_n\}$ for $V$. Define $T: V \to F^n$ by $T(a_1 v_1 + \cdots + a_n v_n) = (a_1, \ldots, a_n)$. This map is linear, injective (kernel = {0}), and surjective (any tuple has a preimage). $\blacksquare$

Consequences:

• All $n$-dimensional vector spaces over the same field are isomorphic to each other.
• Isomorphic spaces share all vector-space properties (dimension, subspace structure, etc.).
• This justifies working in $F^n$ as a "universal model" for any finite-dimensional space.

Example: The space $P_2$ of polynomials of degree at most 2 is isomorphic to $\mathbb{R}^3$ via $a + bx + cx^2 \mapsto (a, b, c)$.