Change of Basis

Section 4.6 Change of Basis

Goals.

We will

consider the affect of a basis on coordinates, and
discover how to convert coordinates from one basis to another,
construct the change of basis matrix, and
use the change of basis matrix to change coordinates from one basis to another.

We have learned to express vectors as linear combinations of basis vectors. We call the coefficients of these linear combinations the coordinates with respect to that basis. We also have seen that a basis is not unique. This means the same vector has different coordinates with respect to different bases. Here we learn to move from one basis to another.

Subsection 4.6.1 Method for Changing Basis

To introduce change of basis we will start by calculating coordinates with respect to two bases and then discovering how we can convert.

Activity 4.6.1.

\(\B=\{x^2,x,1\}\) and \(\C=\{x^2+1,x^2+x,x+1\}\) are both bases for the vector space of all polynomials of degree two or less.

(a)

Find the coordinates of the following vectors with respect to \(\B\text{.}\) \(x^2+x+5,\) \(x^2+5x+8,\) \(x^2+1,\) \(x^2+x,\) \(x+1\)

(b)

Find the coordinates of the following vectors with respect to \(\C\text{.}\) \(x^2+x+5,\) \(x^2+5x+8,\) \(x^2,\) \(x,\) \(1\)

(c)

Write out \(x^2+x+5=a_1\vec{b}_1+a_2\vec{b}_2+a_3\vec{b}_3\) (as a linear combination of \(\B\)).

(d)

Substitute \(\vec{b}_i=r_1\vec{c}_1+r_2\vec{c}_2+r_3\vec{c}_3\) (\(\vec{b}_i\) as linear combinations of \(\C\)) into the previous step.

(e)

Collect so it looks like \(x^2+x+5=q_1\vec{c}_1+q_2\vec{c}_2+q_3\vec{c}_3\text{.}\) Do not simplify the coefficients.

(f)

As a separate step simplify the coefficients.

(g)

Compare this result to \([x^2+x+5]_{\C}\text{.}\)

This illustrates how we can convert a coordinate from one basis into a coordinate for a different basis. Note how it depends on calculating coordinates of the coordinates.

Activity 4.6.2.

We can use the calculations from the previous activity to recognize the simple way to convert coordinates with respect to one basis into coordinates with respect to a different basis.

(a)

Consider the unsimplified coefficients (Task 4.6.1.e). This is the result of matrix arithmetic, which operation?

(b)

Write this matrix operation (write the matrix and vector).

(c)

What is in each column of the matrix?

(d)

What is the vector?

Subsection 4.6.2 Executing Change of Basis

Let’s practice using this technique.

\begin{align*} \left[ \begin{array}{*{3}{r}|*{3}{r}} 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 \\ \end{array} \right] \sim & \begin{array}{l} \\ \\ R_3 \leftarrow -1R_1+R_3 \\ \end{array}\\ \left[ \begin{array}{*{3}{r}|*{3}{r}} 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & -1 & 1 & -1 & 0 & 1 \\ \end{array} \right] \sim & \begin{array}{l} \\ \\ R_3 \leftarrow 1R_2+R_3 \\ \end{array}\\ \left[ \begin{array}{*{3}{r}|*{3}{r}} 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 2 & -1 & 1 & 1 \\ \end{array} \right] \sim & \begin{array}{l} R_1 \leftarrow -1R_2+R_1 \\ \\ \\ \end{array}\\ \left[ \begin{array}{*{3}{r}|*{3}{r}} 1 & 0 & -1 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 2 & -1 & 1 & 1 \\ \end{array} \right] \sim & \begin{array}{l} R_1 \leftarrow \frac{1}{2}R_3+R_1 \\ R_2 \leftarrow -\frac{1}{2}R_3+R_2 \\ R_3\frac{1}{2} \leftarrow R_3 \\ \end{array}\\ \left[ \begin{array}{*{3}{r}|*{3}{r}} 1 & 0 & 0 & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\ 0 & 0 & 1 & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \end{array} \right] \end{align*}

Subsection 4.6.3 Definition of Change of Basis Matrix

For bases \(\B=\{\vec{b}_1, \vec{b}_2, \vec{b}_3\}\) and \(\C=\{\vec{c}_1, \vec{c}_2, \vec{c}_3\}\) of a vector space \(V\)

\begin{equation*} [\vec{x}]_{\C} = \underset{\C \leftarrow \B}{P} [\vec{x}]_{\B} \end{equation*}

where \(\underset{\C \leftarrow \B}{P} = [ [\vec{b}_1]_{\C}, [\vec{b}_2]_{\C}, \ldots, [\vec{b}_n]_{\C}]. \)

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