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9: Change of Basis

Last updated

Jul 25, 2023
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- 8.11E: An Application to Statistical Principal Component Analysis Exercises
- 9.1: The Matrix of a Linear Transformation

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If $A$ is an $m \times n$ matrix, the corresponding matrix transformation $T_A : \mathbb{R}^n \to \mathbb{R}^m$ is defined by

$T_A (\mathbf{x}) = A \mathbf{x} \quad \mbox{ for all columns } \mathbf{x} \mbox{ in } \mathbb{R}^n \nonumber$

It was shown in Theorem [thm:005789] that every linear transformation $T : \mathbb{R}^n \to \mathbb{R}^m$ is a matrix transformation; that is, $T = T_A$ for some $m \times n$ matrix $A$ . Furthermore, the matrix $A$ is uniquely determined by $T$ . In fact, $A$ is given in terms of its columns by

$A = \left[ \begin{array}{cccc} T(\mathbf{e}_1) & T(\mathbf{e}_2) & \cdots & T(\mathbf{e}_n) \end{array}\right] \nonumber$

where $\{\mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n\}$ is the standard basis of $\mathbb{R}^n$ .

In this chapter we show how to associate a matrix with any linear transformation $T : V \to W$ where $V$ and $W$ are finite-dimensional vector spaces, and we describe how the matrix can be used to compute $T(\mathbf{v})$ for any $\mathbf{v}$ in $V$ . The matrix depends on the choice of a basis $B$ in $V$ and a basis $D$ in $W$ , and is denoted $M_{DB}(T)$ . The case when $W = V$ is particularly important. If $B$ and $D$ are two bases of $V$ , we show that the matrices $M_{BB}(T)$ and $M_{DD}(T)$ are similar, that is $M_{DD}(T) = P^{-1}M_{BB}(T)P$ for some invertible matrix $P$ . Moreover, we give an explicit method for constructing $P$ depending only on the bases $B$ and $D$ . This leads to some of the most important theorems in linear algebra, as we shall see in Chapter [chap:11].

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