📂Linear Algebra

Linear Transformation Spaces and Their Matrix Representation Spaces are Isomorphic

Theorem¹

Let’s assume that two vector spaces $V, W$ have dimensions $n, m$, respectively. And let $\beta, \gamma$ be the ordered bases for each. Then the function defined as follows $\Phi$ is an isomorphism.

$$\Phi : L(V, W) \to M_{m\times n}(\mathbb{R}) \quad \text{ by } \quad \Phi (T) = [T]_{\beta}^{\gamma}$$

$[T]_{\beta}^{\gamma}$ is the matrix representation of $T$.

Corollary

A necessary and sufficient condition for a linear transformation to be an isomorphism is that the dimensions of the domain and codomain are equal, hence,

$$\dim( L(V,W) ) = \dim (M_{m\times n}) = mn = \dim(V) \dim(W)$$

Description

In the case of finite dimensions, every linear transformation corresponds to a matrix and vice versa. Addition and composition (matrix multiplication for matrices) are well-preserved between the two, and thus one can think of linear transformations and matrices as being essentially the same. Applying $T$ to the elements of $V$ (the left below) or multiplying the coordinate vector by the matrix representation (the right below) are essentially the same.

$$\mathbf{w} = T(\mathbf{v}) \qquad\qquad[\mathbf{w}]_{\gamma} = [T]_{\beta}^{\gamma} [\mathbf{x}]_{\beta}$$

Proof

First, that $\Phi$ is a linear transformation can be easily seen as follows.

$\Phi (aT + U) = \href{../3283}{[aT+U]_{\beta}^{\gamma} = a[T]_{\beta}^{\gamma} + [U]_{\beta}^{\gamma}} = a\Phi (T) + \Phi (U)$

Now, to show that $\Phi$ is both injective and surjective, let us say $\beta = \left\{ \mathbf{v}_{1}, \dots, \mathbf{v}_{n} \right\}, \gamma = \left\{ \mathbf{w}_{1}, \dots, \mathbf{w}_{n} \right\}$. And let a matrix $A \in M_{m\times n}(\mathbb{R})$ for any given $m \times n$ be provided. Then there exists a linear transformation $T : V \to W$ that satisfies the following uniquely.

$$T(\mathbf{v}_{j}) = \sum_{i=1}^{n}A_{ij}\mathbf{w}_{i},\quad \text{ for } 1\le j \le n$$

This implies $[T]_{\beta}^{\gamma} = A$, in other words, $\Phi (T) = A$. Hence, $T \in L(V, W)$ is uniquely determined for all $A \in M_{m\times n}(\mathbb{R})$, therefore $\Phi$ is a bijection and an isomorphism.

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