Matrix Representations of Linear Transformations

Theorem: Matrix Representation of a Linear Transformation

Every linear transformation $T : V \to W$ from the vector space $(V, F, +, \cdot)$ in the vector space $(W, F, +, \cdot)$ can be represented as a matrix $M_{T} \in F^{d i m (W) \times d i m (V)}$ .

Let $B_{V}$ and $B_{W}$ be ordered bases of $V$ and $W$ , respectively. For every $v \in V$ , we have
$_{B_{W}} T (v) = M_{T} \cdot_{B_{V}} v,$
where $_{B_{V}} v$ is the coordinate vector of $v$ with respect to $B_{V}$ and $_{B_{W}} T (v)$ is the coordinate vector with respect to $B_{W}$ of the vector $T (v) \in W$ which is the result of applying $T$ to $v$ .

Warning: Dependence on the Choice of Bases

The matrix $M_{T}$ depends on the choice of $B_{V}$ and $B_{W}$ - different bases will make the coefficients of $M_{T}$ different. To make this clear, we usually denote $M_{T}$ as $_{B_{W}} [M_{T}]_{B_{V}}$ .

Algorithm: Finding the Matrix Representation

Let $(V, F, +, \cdot)$ and $(W, F, +, \cdot)$ be two vector spaces and let $T : V \to W$ be a linear transformation from $V$ to $W$ .

We want to find the matrix representation $_{B_{W}} [M_{T}]$ with respect to the bases of our choice - $B_{V} = (b_{1}^{V}, \dots, b_{m}^{V})$ for $V$ and $B_{W} = (b_{1}^{W}, \dots, b_{n}^{W})$ for $W$ .

Determine the effect of $T$ on the elements of $B_{V}$ , i.e.

$w_{1} = T (b_{1}^{V}) \dots w_{m} = T (b_{m}^{V})$

Determine the coordinate vector $_{B_{W}} w_{1}, \dots,_{B_{W}} w_{m}$ of $w_{1}, \dots, w_{n}$ with respect to $B_{W}$ .

There is no general process for this - it is different for each vector space.

Construct $_{B_{W}} [M_{T}]$ by using the coordinate vectors $_{B_{W}} w_{1}, \dots,_{B_{W}} w_{m}$ as its columns:

$_{B_{W}} [M_{T}]_{B_{V}} = ∣_{B_{W}} w_{1} ∣ ∣ \dots ∣ ∣_{B_{W}} w_{m} ∣ = ∣_{B_{W}} T (b_{1}^{V}) ∣ ∣ \dots ∣ ∣_{B_{W}} T (b_{m}^{V}) ∣$

EXAMPLE

TODO

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Matrix Representations of Linear Transformations

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