```
(
[ 1 0 1 1] [ 1 0 0 2]
[ 1 1 0 5] [ 0 1 0 3]
[ 1 0 2 0] [ 0 0 1 -1]
[ 1 -1 0 -1], [ 0 0 0 0]
)
```

Math 303: Section 25

Dr. Janssen

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

**Takeaway:** Bases give coordinate systems!

**Definition 1 **Let \(\B = \set{\v_1, \v_2, \ldots, \v_n}\) be a basis for a vector space \(V\). For any vector \(\x\) in \(V\), the **coordinate vector of \(\x\) with respect to \(\B\) is the vector

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

where

\[ \x = x_1 \v_1 + x_2 \v_2 + \cdots + x_n \v_n. \]

The scalars \(x_1, x_2, \ldots, x_n\) are the **coordinates of the vector \(\x\) with respect to the basis**.

Let \(\mathcal{S} = \set{1,t}\) and \(\B = \set{3+2t, 1-t}\). Assume that \(\mathcal{S}\) and \(\B\) are bases for \(\P_1\). Find \([3+7t]_{\mathcal{S}}\) and \([3+7t]_{\B}\).

**Definition 2 **Let \(V\) be an \(n\)-dimensional vector space with basis \(\B\). The **coordinate transformation** \(T\) from \(V\) to \(\R^n\) with respect to the basis \(\B\) is the mapping defined by

\[ T(\x) = [\x]_{\B} \]

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

Let \(a(t) = 8 + 2t\) and \(b(t) = -5 + t\) in \(\P_1\). Suppose we know that \(\C = \set{1+t, 2-t}\) is a basis of \(\P_1\). Let \(T(\x) = [\x]_{\C}\) for \(\x\) in \(\P_1\).

- What are \(T(a(t))\) and \(T(b(t))\)?
- Find \(T(a(t)) + T(b(t))\).
- What is \(T(a(t) + b(t)) = [a(t) + b(t)]_{\C}\)?
- What is the relationship between \(T(a(t)) + T(b(t))\) and \(T(a(t) + b(t))\)?
- Show that if \(c\) is any scalar, then \(T(ca(t)) = c T (a(t))\).
- Where have we seen functions with these properties before?

**Theorem 1 **If a vector space \(V\) has a basis \(\B\) of \(n\) vectors, then the coordinate mapping \(T : V\to \R^n\) defined by \(T(\x) = [\x]_{\B}\) satisfies

- \(T(\u + \w) = T(\u) + T(\w)\) and
- \(T(c\u) = c T(\u)\)

Let \(V\) be a vector space with an ordered basis \(\B = \set{\v_1, \v_2, \ldots, \v_n}\). Then \(T\) maps \(V\) into \(\R^n\). Let’s show that \(T\) maps \(V\) *onto* \(\R^n\).

Let \(\mathbf{b} = [b_1 \ b_2 \ \cdots \ b_n]^{\textsf{T}}\) be a vector in \(\R^n\). Must there be a vector \(\v\) in \(V\) so that \(T(\v) = \mathbf{b}\)? If so, find such a vector. If not, explain why not.

**Theorem 2 **If a vector space \(V\) has a basis \(\B\) of \(n\) vectors, then the coordinate mapping \(T : V\to \R^n\) defined by \(T(\x) = [\x]_{\B}\) is both one-to-one and onto.

Let \(V = \P_3\) and let \(\B = \set{1, t, t^2, t^3}\). Let \(S = \set{1+t+t^2+t^3, t - t^3, 1+ 2t^2, 1+ 5t - t^3}\).

- Find each of \([1+t+t^2+t^3]_{\B}, [t - t^3]_{\B}, [1+2t^2]_{\B}\), and \([1+5t - t^3]_{\B}\).
- Are the vectors \([1+t+t^2+t^3]_{\B}, [t - t^3]_{\B}, [1+2t^2]_{\B}\), and \([1+5t - t^3]_{\B}\) linearly independent or dependent? Explain. If the vectors are linearly independent, write one of the vectors as a linear combination of the others.
- The coordinate transformation identifies the vectors in \(S = \set{1+t+t^2+t^3, t - t^3, 1+ 2t^2, 1+ 5t - t^3}\) with their coordinate vectors in \(\R^4\). Use that information to determine if \(S\) is a linearly independent or dependent set. If dependent, write one of the vectors in \(S\) as a linear combination of the others.

`sage`

```
(
[ 1 0 1 1] [ 1 0 0 2]
[ 1 1 0 5] [ 0 1 0 3]
[ 1 0 2 0] [ 0 0 1 -1]
[ 1 -1 0 -1], [ 0 0 0 0]
)
```