23. Coordinate System

Dated: 03-06-2025

Theorem

Let \(B =\{\vec b_1, \vec b_2, \ldots, \vec b_n\}\) be a basis for a vector space¹ \(\mathbb V\). Then

\[\vec x = c_1 \vec b_1, \, c_2 \vec b_2, \ldots, \, c_n \vec b_n \quad \forall \vec x \in \mathbb V\]

Definition (\(B\) Coordinate of \(\vec x\))

The \(c_1, c_2, \ldots, c_n\) are the scalar weights, called the \(B\) coordinates of \(\vec x\).

Example

Let \(S = \{\vec v_1, \vec v_2, \vec v_3\}\) be the basis for \(\mathbb R^3\) where \(\vec v_1 = \langle1, 2, 1\rangle\), \(\vec v_2 = \langle2, 9, 0\rangle\) and \(\vec v_3 = \langle3, 3, 4\rangle\)

Find the coordinate vector of \(\vec v = \langle5, -1, 9\rangle\) with respect to \(S\).
Find the vector² \(\vec v \in \mathbb R^3\) whose coordinate vector with respect to the basis \(S\) is \([\vec v]_S = \langle-1, 3, 2\rangle\)

Solution

Part 1

\[\because \vec x = c_1 \vec v_1 + c_2 \vec v_2 + c_3 \vec v_3\]

\[\langle5, -1, 9\rangle = c_1 \langle1, 2, 1\rangle + c_2\langle2, 9, 0\rangle + c_3 \langle3, 3, 4\rangle\]

\[ \begin{array}{cccccccccc} & c_1 &+ &2& c_2 &+ &3& c_3 &= &5\\ 2 &c_1 &+ &9&c_2 &+ &3&c_3 &= &-1\\ &c_1 &&&&+ &4&c_3 &= &9\\ \end{array} \]

After solving this linear system,³ we have

\[c_1 = 1, c_2 = -1, c_3 = 2\]

\[\implies [\vec v]_S = \langle1, -1, 2\rangle\]

Part 2

\[[\vec v]_S = \langle-1, 3, 2\rangle\]

\(\(c_1 = -1, c_2 = 3, c_3 = 2\)\)

Plugging it into the original equation, we have

\[(-1) \langle1, 2, 1\rangle + (3)\langle2, 9, 0\rangle + (2)\langle3, 3, 4\rangle\]

\[\vec v= \langle11, 31, 7\rangle\]

Change-of-coordinates Matrix

\[\vec x = c_1 \vec b_1 + c_2 \vec b_2 + \ldots + c_n \vec b_n\]

\[ \vec x = \begin{bmatrix} \vec b_1 & \vec b_2 & \cdots & \vec b_n \end{bmatrix} \begin{bmatrix} c_1\\ c_2\\ \vdots\\ c_n \end{bmatrix} = P_B[\vec x]_B \]

The vector¹ \(P_B\) is called the change of coordinates matrix from \(B\) to standard basis of \(\mathbb R^n\).

Coordinate Mapping

Let \(B = \{\vec b_1, \vec b_2, \ldots, b_n\}\) be a basis for a vector space¹ \(\mathbb V\). Then the mapping \(\vec x \to [\vec x]_B\) is a one to one linear transformation⁴ from \(\mathbb V\) onto \(\mathbb R^n\).
This transformation⁴ is called isomorphism(iso meaning "same" and morph meaning "form").