Change of Basis

Math 303: Section 26

Dr. Janssen

Intro

\[ \def\R{{\mathbb R}} \def\P{{\mathbb P}} \def\B{{\mathcal B}} \def\C{{\mathcal C}} \def\b{{\mathbf{b}}} \def\c{{\mathbf{c}}} \def\x{{\mathbf{x}}} \def\y{{\mathbf{y}}} \def\u{{\mathbf{u}}} \def\v{{\mathbf{v}}} \def\w{{\mathbf{w}}} \def\z{{\mathbf{z}}} \def\M{{\mathcal{M}}} \DeclareMathOperator{\null}{Nul} \DeclareMathOperator{\span}{Span} \DeclareMathOperator{\dim}{dim} \newcommand{\set}[1]{\left\{ {#1} \right\}} \newcommand{\setof}[2]{{\left\{#1\,\colon\,#2\right\}}} \]

Preview Activity 26.1 Discussion

Takeaway: We can change bases¹ via coordinate vectors!

1. Even for non-\(\R^n\) spaces!

Motivation

Let \(\B = \set{\b_1, \b_2, \ldots, \b_n}\) and \(\C = \set{\c_1, \c_2, \ldots, \c_n}\) be two bases for a vector space \(V\). If \(\x\) is in \(V\), we can write

\[ \x = x_1 \b_1 + x_2 \b_2 + \cdots + x_n \b_n. \]

Then

\[ [\x]_\B = \left[\begin{matrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{matrix}\right]. \]

Observe

\[ \begin{align*} [\x]_\C &= [x_1 \b_1 + x_2 \b_2 + \cdots + x_n \b_n]_\C \\ &= x_1 [\b_1]_\C \end{align*} \]