# Chapter 23. Coordinate Systems # Chapter 23. Coordinate Systems A coordinate system is a way to describe vectors by scalars. In linear algebra, coordinates are not attached to a vector by nature. They depend on a chosen basis. A vector is an element of a vector space. A coordinate vector is a column of scalars that represents that vector relative to an ordered basis. The same vector can have different coordinate vectors under different bases. A basis gives each vector a unique coordinate representation. ## 23.1 Coordinates from a Basis Let \(V\) be a vector space over a field \(F\). Let $$ B=(v_1,v_2,\ldots,v_n) $$ be an ordered basis of \(V\). Since \(B\) spans \(V\), every vector \(v\in V\) can be written as $$ v=c_1v_1+c_2v_2+\cdots+c_nv_n. $$ Since \(B\) is linearly independent, the scalars $$ c_1,c_2,\ldots,c_n $$ are unique. The coordinate vector of \(v\) with respect to \(B\) is $$ [v]_B= \begin{bmatrix} c_1\\ c_2\\ \vdots\\ c_n \end{bmatrix}. $$ The notation \([v]_B\) means “the coordinates of \(v\) in the basis \(B\).” ## 23.2 Ordered Bases A coordinate system requires an ordered basis. The set $$ \{v_1,v_2\} $$ and the set $$ \{v_2,v_1\} $$ contain the same vectors, but they give different coordinate columns. If $$ v=3v_1+5v_2, $$ then with $$ B=(v_1,v_2) $$ we have $$ [v]_B= \begin{bmatrix} 3\\ 5 \end{bmatrix}. $$ With $$ C=(v_2,v_1), $$ the same vector is written as $$ v=5v_2+3v_1, $$ so $$ [v]_C= \begin{bmatrix} 5\\ 3 \end{bmatrix}. $$ The order of the basis determines the order of the coordinates. ## 23.3 Standard Coordinates in \(\mathbb{R}^n\) The standard basis of \(\mathbb{R}^n\) is $$ E=(e_1,e_2,\ldots,e_n), $$ where $$ e_1= \begin{bmatrix} 1\\ 0\\ \vdots\\ 0 \end{bmatrix}, \quad e_2= \begin{bmatrix} 0\\ 1\\ \vdots\\ 0 \end{bmatrix}, \quad \ldots, \quad e_n= \begin{bmatrix} 0\\ 0\\ \vdots\\ 1 \end{bmatrix}. $$ Every vector $$ x= \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix} $$ has the expansion $$ x=x_1e_1+x_2e_2+\cdots+x_ne_n. $$ Therefore $$ [x]_E= \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix}. $$ In the standard basis, the coordinate vector looks exactly like the vector. This is why it is easy to forget that coordinates depend on a basis. ## 23.4 A Nonstandard Coordinate System Consider the basis $$ B= \left( \begin{bmatrix} 1\\ 1 \end{bmatrix}, \begin{bmatrix} 1\\ -1 \end{bmatrix} \right) $$ of \(\mathbb{R}^2\). Let $$ x= \begin{bmatrix} 7\\ 3 \end{bmatrix}. $$ To find \([x]_B\), solve $$ x=c_1 \begin{bmatrix} 1\\ 1 \end{bmatrix} + c_2 \begin{bmatrix} 1\\ -1 \end{bmatrix}. $$ This gives $$ \begin{bmatrix} 7\\ 3 \end{bmatrix} = \begin{bmatrix} c_1+c_2\\ c_1-c_2 \end{bmatrix}. $$ Thus $$ c_1+c_2=7, \qquad c_1-c_2=3. $$ Adding the equations gives $$ 2c_1=10, $$ so $$ c_1=5. $$ Then $$ c_2=2. $$ Therefore $$ [x]_B= \begin{bmatrix} 5\\ 2 \end{bmatrix}. $$ The vector has standard coordinates \((7,3)\), but its coordinates in \(B\) are \((5,2)\). ## 23.5 Coordinate Maps Given an ordered basis $$ B=(v_1,\ldots,v_n) $$ of \(V\), the coordinate map is the function $$ \Phi_B:V\to F^n $$ defined by $$ \Phi_B(v)=[v]_B. $$ This map sends an abstract vector to its coordinate column. It is linear. If $$ [u]_B= \begin{bmatrix} a_1\\ \vdots\\ a_n \end{bmatrix}, \qquad [v]_B= \begin{bmatrix} b_1\\ \vdots\\ b_n \end{bmatrix}, $$ then $$ u+v=(a_1+b_1)v_1+\cdots+(a_n+b_n)v_n. $$ Therefore $$ [u+v]_B=[u]_B+[v]_B. $$ Similarly, $$ [cu]_B=c[u]_B. $$ Thus coordinates preserve the vector space operations. ## 23.6 Coordinates and Isomorphism The coordinate map $$ \Phi_B:V\to F^n $$ is an isomorphism. It is one-to-one because each vector has unique coordinates. It is onto because every column $$ \begin{bmatrix} c_1\\ \vdots\\ c_n \end{bmatrix} \in F^n $$ comes from the vector $$ c_1v_1+\cdots+c_nv_n. $$ Thus every \(n\)-dimensional vector space over \(F\) can be identified with \(F^n\) after choosing a basis. This identification is useful, but it should be read carefully. The space \(V\) and the coordinate space \(F^n\) are structurally equivalent, but the equivalence depends on the chosen basis. ## 23.7 Coordinate Matrix of a Basis Let $$ B=(v_1,\ldots,v_n) $$ be a basis of \(\mathbb{R}^n\). Place the basis vectors as columns of a matrix: $$ P_B= \begin{bmatrix} |&|&&|\\ v_1&v_2&\cdots&v_n\\ |&|&&| \end{bmatrix}. $$ Then for any vector \(x\in\mathbb{R}^n\), $$ x=P_B[x]_B. $$ This equation converts \(B\)-coordinates into standard coordinates. The matrix \(P_B\) is invertible because its columns form a basis. Therefore $$ [x]_B=P_B^{-1}x. $$ This equation converts standard coordinates into \(B\)-coordinates. ## 23.8 Example Using a Basis Matrix Let $$ B= \left( \begin{bmatrix} 1\\ 1 \end{bmatrix}, \begin{bmatrix} 1\\ -1 \end{bmatrix} \right). $$ Then $$ P_B= \begin{bmatrix} 1&1\\ 1&-1 \end{bmatrix}. $$ For $$ [x]_B= \begin{bmatrix} 5\\ 2 \end{bmatrix}, $$ standard coordinates are $$ x=P_B[x]_B = \begin{bmatrix} 1&1\\ 1&-1 \end{bmatrix} \begin{bmatrix} 5\\ 2 \end{bmatrix} = \begin{bmatrix} 7\\ 3 \end{bmatrix}. $$ Conversely, if $$ x= \begin{bmatrix} 7\\ 3 \end{bmatrix}, $$ then $$ [x]_B=P_B^{-1}x. $$ Since $$ P_B^{-1} = \frac{1}{-2} \begin{bmatrix} -1&-1\\ -1&1 \end{bmatrix} = \begin{bmatrix} 1/2&1/2\\ 1/2&-1/2 \end{bmatrix}, $$ we get $$ [x]_B= \begin{bmatrix} 1/2&1/2\\ 1/2&-1/2 \end{bmatrix} \begin{bmatrix} 7\\ 3 \end{bmatrix} = \begin{bmatrix} 5\\ 2 \end{bmatrix}. $$ ## 23.9 Change of Coordinates A change of coordinates converts the coordinate vector of the same vector from one basis to another. Let $$ B=(b_1,\ldots,b_n) $$ and $$ C=(c_1,\ldots,c_n) $$ be bases of \(V\). For a vector \(v\), one may know $$ [v]_B $$ and want $$ [v]_C. $$ The vector \(v\) itself does not change. Only its coordinate description changes. In \(\mathbb{R}^n\), if \(P_B\) and \(P_C\) are the basis matrices, then $$ v=P_B[v]_B $$ and $$ v=P_C[v]_C. $$ Thus $$ P_B[v]_B=P_C[v]_C. $$ Multiplying by \(P_C^{-1}\), $$ [v]_C=P_C^{-1}P_B[v]_B. $$ The matrix $$ P_{C\leftarrow B}=P_C^{-1}P_B $$ is called the change-of-coordinates matrix from \(B\)-coordinates to \(C\)-coordinates. ## 23.10 Identity Map, Different Coordinates The formula $$ [v]_C=P_C^{-1}P_B[v]_B $$ represents the identity map on \(V\), not a transformation of the vector itself. The vector remains \(v\). The coordinate column changes because the basis changes. This distinction is important. A matrix can represent either: | Matrix role | Meaning | |---|---| | Linear transformation | Changes the vector | | Change of coordinates | Changes the description of the same vector | The same numerical matrix may appear in either role, but the interpretation differs. ## 23.11 Coordinates in Polynomial Spaces Let $$ P_2 $$ be the vector space of polynomials of degree at most \(2\). Let $$ B=(1,x,x^2). $$ For $$ p(x)=3-5x+2x^2, $$ the coordinate vector is $$ [p]_B= \begin{bmatrix} 3\\ -5\\ 2 \end{bmatrix}. $$ Now choose another basis: $$ C=(1,1+x,1+x+x^2). $$ To find \([p]_C\), solve $$ p(x)=a\cdot 1+b(1+x)+c(1+x+x^2). $$ The right-hand side is $$ a+b+c+(b+c)x+cx^2. $$ Match coefficients with $$ 3-5x+2x^2. $$ Then $$ c=2, $$ $$ b+c=-5, $$ so $$ b=-7, $$ and $$ a+b+c=3, $$ so $$ a=8. $$ Thus $$ [p]_C= \begin{bmatrix} 8\\ -7\\ 2 \end{bmatrix}. $$ The polynomial has not changed. Its coordinates changed because the basis changed. ## 23.12 Coordinates in Matrix Spaces Let $$ M_{2\times 2}(\mathbb{R}) $$ have the ordered basis $$ B=(E_{11},E_{12},E_{21},E_{22}), $$ where $$ E_{11}= \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}, \quad E_{12}= \begin{bmatrix} 0&1\\ 0&0 \end{bmatrix}, $$ $$ E_{21}= \begin{bmatrix} 0&0\\ 1&0 \end{bmatrix}, \quad E_{22}= \begin{bmatrix} 0&0\\ 0&1 \end{bmatrix}. $$ For $$ A= \begin{bmatrix} 2&-1\\ 7&4 \end{bmatrix}, $$ we have $$ A=2E_{11}-E_{12}+7E_{21}+4E_{22}. $$ Therefore $$ [A]_B= \begin{bmatrix} 2\\ -1\\ 7\\ 4 \end{bmatrix}. $$ A matrix can itself be a vector. Its coordinates are determined by the ordered basis chosen for the matrix space. ## 23.13 Coordinates in Function Spaces Let $$ W=\operatorname{span}(1,\cos x,\sin x) $$ with ordered basis $$ B=(1,\cos x,\sin x). $$ For $$ f(x)=4-2\cos x+5\sin x, $$ the coordinate vector is $$ [f]_B= \begin{bmatrix} 4\\ -2\\ 5 \end{bmatrix}. $$ Again, coordinates record coefficients relative to a basis. The vector may be a function, a polynomial, a matrix, or another object. The coordinate principle is the same. ## 23.14 Coordinate Systems and Linear Equations Coordinate systems convert vector equations into scalar equations. Suppose $$ v=c_1v_1+\cdots+c_nv_n. $$ Finding the coordinates of \(v\) means solving for the unknown scalars \(c_1,\ldots,c_n\). In \(\mathbb{R}^n\), this becomes the linear system $$ P_Bc=v, $$ where $$ c=[v]_B. $$ Because \(P_B\) is invertible, the system has a unique solution. Thus coordinate computation is a special case of solving a linear system. ## 23.15 Coordinate Systems and Geometry In geometry, a coordinate system assigns numbers to points or vectors. In linear algebra, the origin and basis vectors determine the coordinate system for a vector space. For vector spaces, the origin is fixed as the zero vector. A basis determines the coordinate axes. In \(\mathbb{R}^2\), the standard basis gives the usual horizontal and vertical axes. A nonstandard basis gives oblique axes. Coordinates then measure how far to move along each basis direction. If the basis vectors are perpendicular and have length \(1\), the coordinate system is orthonormal. If they are not perpendicular, the coordinate system is oblique. Both are valid linear coordinate systems as long as the basis vectors are linearly independent and span the space. ## 23.16 Coordinate Systems and Computation Coordinates are essential for computation because computers manipulate arrays of numbers. An abstract vector space becomes computationally usable after choosing a basis. Polynomials become coefficient arrays. Matrices become flattened arrays. Functions in finite-dimensional subspaces become coefficient vectors. For example, the polynomial $$ p(x)=a_0+a_1x+\cdots+a_nx^n $$ can be stored as $$ \begin{bmatrix} a_0\\ a_1\\ \vdots\\ a_n \end{bmatrix}. $$ This coordinate representation allows addition, scalar multiplication, differentiation, integration, and evaluation to be implemented as operations on arrays. ## 23.17 Coordinate Dependence Some quantities depend on the chosen coordinate system. Others do not. The coordinate vector \([v]_B\) depends on \(B\). The vector \(v\) itself does not. The matrix of a linear transformation depends on the chosen bases. The transformation itself does not. Dimension does not depend on the basis. Linear independence of an actual list of vectors does not depend on coordinates, although coordinates may be used to test it. This distinction leads to an important habit: separate objects from their representations. ## 23.18 Standard Coordinates Are Not Intrinsic In \(\mathbb{R}^n\), the standard coordinates are so familiar that they may seem intrinsic. They are not. They are coordinates relative to the standard basis. The vector $$ \begin{bmatrix} 2\\ 3 \end{bmatrix} $$ really means $$ 2e_1+3e_2 $$ with respect to the standard basis. Under a different basis, the same vector may have coordinates $$ \begin{bmatrix} 5\\ -1 \end{bmatrix} $$ or another column. The column alone does not fully describe the vector unless the basis is known. A coordinate vector always has an implicit basis. ## 23.19 Common Notation | Notation | Meaning | |---|---| | \(B=(v_1,\ldots,v_n)\) | Ordered basis | | \([v]_B\) | Coordinate vector of \(v\) in basis \(B\) | | \(P_B\) | Basis matrix with columns \(v_1,\ldots,v_n\) | | \(x=P_B[x]_B\) | Convert \(B\)-coordinates to standard coordinates | | \([x]_B=P_B^{-1}x\) | Convert standard coordinates to \(B\)-coordinates | | \(P_{C\leftarrow B}\) | Change-of-coordinates matrix from \(B\) to \(C\) | | \([v]_C=P_{C\leftarrow B}[v]_B\) | Convert coordinates from \(B\) to \(C\) | ## 23.20 Summary A coordinate system is determined by an ordered basis. It represents each vector by the unique scalars needed to express that vector as a linear combination of the basis vectors. The key ideas are: | Concept | Meaning | |---|---| | Coordinate vector | Coefficients of a vector relative to a basis | | Ordered basis | Basis with fixed order | | Coordinate map | Map \(v\mapsto [v]_B\) | | Basis matrix | Matrix whose columns are basis vectors | | Change of coordinates | Conversion between coordinate descriptions | | Standard coordinates | Coordinates relative to the standard basis | | Coordinate dependence | Representations depend on basis | | Intrinsic object | The vector itself, independent of coordinates | Coordinates make vector spaces computable. They turn abstract vectors into columns of scalars. But coordinates are representations, not the underlying objects. The basis is part of the meaning.