Isomorphisms

Say that $V$ isomorphic to $W$ if there exists a linear transformation $T:V\to W$ (called an isomorphism) that is invertible

Facts

• $V$, $W$ finite-dim vector spaces. $V$ isomorphic to $W$ iff $\dim (V)=\dim (W)$

Useful Isomorphisms

• Linear Transformations and Matrices
  • $V,W$ finite-dim vector spaces over F. $\dim (V)=n,\dim (W)=m$
  • Isomorphism: $\Phi :\mathcal{L}(V,W)\to M_{m\times n}(F)$ defined by $\Phi (T)=[T{]}_{\beta }^{\gamma }$
  • Corollary: $\mathcal{L}(V,W)$ is finite dimensional of dimension $m\cdot n$
• Standard Representation of $V$ with respect to $\beta$
  • $\beta$ ordered basis for $V$ with $\dim (V)=n$ over $F$
  • Isomorphism: ${\varphi }_{\beta }:V\to F^n$ defined by ${\varphi }_{\beta }(x)=[x{]}_{\beta }$