Isomorphisms and Composition

Chapter 7 • Section 7-3

"When two spaces are effectively the same, just dressed differently. 👯"

🤝 Isomorphisms

An isomorphism is a linear transformation $T: V \to W$ that is both:

  • One-to-One: No information is lost.
  • Onto: The entire target space is covered.

If an isomorphism exists, we say $V \cong W$. They have the same dimension and structure.

Example: $\calP_2 \cong \mathbb{R}^3$

The map $a + bx + cx^2 \mapsto (a, b, c)$ is an isomorphism.

Polynomials act just like 3D vectors!

Example: $\M_{22} \cong \mathbb{R}^4$

Mapping a $2 \times 2$ matrix to a vector of its 4 entries is an isomorphism.

🌐 Isomorphism Theorems

Dimension Theorem

Two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension.

$V \cong W \iff \dim(V) = \dim(W)$

Invertibility Theorem

A linear transformation $T$ is an isomorphism if and only if it is invertible.

$T^{-1}$ is also a linear transformation!

Coordinate Isomorphism

Every vector space $V$ of dimension $n$ is isomorphic to $\mathbb{R}^n$.

If $B = \{\vec{b}_1, \dots, \vec{b}_n\}$ is a basis, the map $C_B: V \to \mathbb{R}^n$ sends a vector to its coordinates:

$$ [\vec{v}]_B = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix} $$

This is why we can solve almost any problem using $\mathbb{R}^n$!

Composition Machine

See how applying two transformations in sequence works. Order matters!
Start with vector $\vec{v} = (1, 0)$.

Result: $T(S(\vec{v}))$
(0, 2)

Step 1: (1, 0) → (0, 1)
Step 2: (0, 1) → (0, 2)

Inverse Transformations

If $T$ is an isomorphism, it has an inverse $T^{-1}$ such that:

$$ T^{-1}(T(\vec{v})) = \vec{v} \quad \text{and} \quad T(T^{-1}(\vec{w})) = \vec{w} $$

It's like an "undo" button for the transformation.

🧠 Knowledge Check Win $15

Is the space of $2 \times 3$ matrices isomorphic to $\mathbb{R}^5$?