Change of Basis

Math 303: Section 26

Dr. Janssen

Intro

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

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

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

The definition

Definition 1 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 \(V\). The change of basis matrix from \(\B\) to \(\C\) is the matrix

\[ \mathop{P}_{\C\leftarrow \B} = \left[ [\b_1]_\C \ [\b_2]_\C \ \cdots \ [\b_n]_\C \right]. \]

The theorem

Theorem 1 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 \(V\). Then

\[ [\x]_\C = \mathop{P}_{\C\leftarrow \B} [\x]_\B \]

for any vector \(\x\) in \(V\).

Finding a Change of Basis Matrix

How to find a cob matrix?

We’ll use the coordinate transformation to leverage the work we did in \(\R^n\).

Activity 26.1

Let \(b_1(t) = 4 + t\), \(b_2(t) = 2 + 5t\), \(c_1(t) = -1 + 2t\), \(c_2(t) = -1 - t\). The sets \(\B = \set{b_1(t), b_2(t)}\) and \(\C = \set{c_1(t), c_2(t)}\) are bases for \(\P_1\). Our goal is to find the change of basis matrix \(\mathop{P}_{\C\leftarrow \B}\) from \(\B\) to \(\C\). We will use the coordinate transformation with respect to the standard basis \(\mathcal{S} = \set{1,t}\) for \(\P_1\) to transfer our work into \(\R^2\).

• What are \([b_1(t)]_\S\), \([b_2(t)]_\S\), \([c_1(t)]_\S\), and \([c_2(t)]_\S\)?
• What matrix equation must we solve to write \([b_1(t)]_\S\) as a linear combination of the vectors \([c_1(t)]_\S\) and \([c_2(t)]_\S\)? How do we solve this equation? (Don’t solve it yet.)
• What matrix equation must we solve to write \([b_2(t)]_\S\) as a linear combination of the vectors \([c_1(t)]_\S\) and \([c_2(t)]_\S\)? How do we solve this equation? (Don’t solve it yet.)
• Let \(A\) be the coefficient matrix of the systems you wrote in parts (b) and (c). Find the reduced row echelon form of the augmented matrix \([A \ | \ [b_1(t)]_\S \ [b_2(t)]_\S]\).
• Use the result of the previous part to write \(b_1(t)\) and \(b_2(t)\) as linear combinations of \(c_1(t)\) and \(c_2(t)\).
• Based on your responses to the second and third parts, if \([I_2 \ | \ P]\) is the reduced row echelon form of the matrix \([A \ | \ [b_1(t)]_\S \ [b_2(t)]_\S]\), what property will the matrix \(P\) have? Explain.

Properties of the Change of Basis Matrix

Activity 26.2

The sets \(\B = \set{3, 4-t}\) and \(\C = \set{1+2t, -1 + t}\) are bases for \(\P_1\).

• Find the change of basis matrix \(\mathop{P}_{\C\leftarrow\B}\) from the basis \(\B\) to the basis \(\C\).
• Let \(p(t) = 2 + 4t\). Find \([p(t)]_\B\) and \([p(t)]_\C\).
• Verify by matrix multiplication that \([p(t)]_\C = \mathop{P}_{\C\leftarrow\B} [p(t)]_\B\).
• Find the change of basis matrix \(\mathop{P}_{\B\leftarrow \C}\) from the basis \(\C\) to the basis \(\B\).
• Verify by matrix multiplication that \([p(t)]_\B = \mathop{P}_{\B\leftarrow\C} [p(t)]_\C\).
• How, specifically, are the matrices \(\mathop{P}_{\C\leftarrow\B}\) and \(\mathop{P}_{\B\leftarrow\C}\) related? (If you don’t see it, multiply them!)

Theorem 26.3

Theorem 2 Let \(V\) be a finite dimensional vector space, and let \(\B, \C\), and \(\S\) be bases for \(V\).

• The change of basis matrix \(\mathop{P}_{\C\leftarrow\B}\) is invertible,
• \(\mathop{P}_{\C\leftarrow\B}^{-1} = \mathop{P}_{\B\leftarrow\C}\)
• \(\mathop{P}_{\S\leftarrow \C} \mathop{P}_{\C\leftarrow\B} = \mathop{P}_{\S\leftarrow\B}\)