Coordinate Vectors and Coordinate Transformations

Math 303: Section 25

Dr. Janssen

Intro

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

Preview Activity 25.1 Discussion

Takeaway: Bases give coordinate systems!

Definition

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.

Activity 25.1

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 \([3+7t]_{\B}\).

Should get \([3 \ 7]^\textsf{T}\) and \([2 \ -3]^\textsf{T}\).

Note that the need to specify coordinates forces us to order our bases! So we should really think of our basis as a sequence of vectors, perhaps denoted with parentheses or langle/rangles.

The Coordinate Transformations

We have observed that \(\P_{n-1}\) “looks like” \(\R^n\) by identifying \(a_0 + a_1 t + \cdots + a_{n-1} t^{n-1}\) with a column of numbers in the standard basis for \(\R^n\).

The transformation from a vector space \(V\) to \(\R^n\) that produces a coordinate vector is called the coordinate transformation.

Definition

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 \(V\).

Activity 25.2

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?

Upshot: coordinate transformations behave like matrix transformations!

Summary: Theorem 25.3

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
• \(T(c\u) = c T(\u)\)

What does it mean for a function to be onto?

What does it mean for a function to be one-to-one?

Activity 25.3

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.

Summary: Theorem 25.4

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.

one-to-one is left as an exercise!

this allows us to identify vector spaces \(V\) with euclidean spaces and transfer questions about eg linear independence, span in \(V\) to \(\R^n\) where we have matrix tools to help answer our questions.

Activity 25.4

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.

A = Matrix([[1,0,1,1],[1,1,0,5],[1,0,2,0],[1,-1,0,-1]])
A.rref()

[ 1  0  0  2]
[ 0  1  0  3]
[ 0  0  1 -1]
[ 0  0  0  0]

In sage

A = Matrix([[1,0,1,1],[1,1,0,5],[1,0,2,0],[1,-1,0,-1]])
A, A.rref()

(
[ 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]
)