Coordinates with Respect to a Basis

In mathematics and physics, we often describe objects like position, force, or even abstract quantities using a set of numbers called coordinates. However, these coordinates are not the object itself; they are merely a description from a particular point of view, or "basis." The significance of this distinction is immense, as the choice of perspective can be the difference between a convoluted problem and an elegant solution. This article addresses a fundamental question in linear algebra: how do we translate our descriptions between different points of view, and why is it so important? We will first explore the core principles in the chapter ​​"Principles and Mechanisms"​​, where we will define what coordinates with respect to a basis truly mean and establish the mechanics for changing them. Following this, the chapter ​​"Applications and Interdisciplinary Connections"​​ will showcase how this abstract machinery becomes a powerful, practical tool in fields ranging from crystallography and computer graphics to the very fabric of spacetime in general relativity.

Imagine you want to describe the location of a statue in a large city square. You might say, "Start at the central fountain, walk 50 meters east, and then 30 meters north." These numbers, (50, 30), are the ​​coordinates​​. They are a set of instructions, a recipe for finding the statue. But this recipe is only meaningful if we all agree on what "east" and "north" mean, and that we're starting from the fountain. "East" and "north" are our fundamental directions; they are our ​​basis​​. Change the starting point or the directions—say, by aligning your map with a diagonal street—and the numbers in your recipe will change completely, even though the statue hasn't moved an inch.

This is the central idea of coordinates in mathematics and physics. A vector—whether it represents a position, a force, or even something as abstract as a polynomial—is a real, definite thing. Its coordinates are merely its shadow, a description of that thing from a particular point of view, or with respect to a particular ​​basis​​. The art and science of linear algebra, in many ways, is about choosing the right perspective to make a problem simple.

Let’s get to the heart of it. When we write a vector in $\mathbb{R}^2$ as $\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$, we're being a bit lazy. We're implicitly saying $\mathbf{v} = v_1 \begin{pmatrix} 1 \\ 0 \end{pmatrix} + v_2 \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. The familiar vectors $\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ form the ​​standard basis​​. They are our default "east" and "north."

But what if we choose a different basis? Suppose we have a new set of basis vectors, say $B = \{\mathbf{b}_1, \mathbf{b}_2\}$ where $\mathbf{b}_1 = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$ and $\mathbf{b}_2 = \begin{pmatrix} -2 \\ 4 \end{pmatrix}$. And suppose a vector $\mathbf{u}$ has the coordinates $\begin{pmatrix} 5 \\ 2 \end{pmatrix}$ with respect to this new basis. What does that mean? It means the recipe to get to $\mathbf{u}$ is "take 5 steps of $\mathbf{b}_1$ and 2 steps of $\mathbf{b}_2$."

Let's follow the recipe: $\mathbf{u} = 5 \mathbf{b}_1 + 2 \mathbf{b}_2 = 5 \begin{pmatrix} 3 \\ -1 \end{pmatrix} + 2 \begin{pmatrix} -2 \\ 4 \end{pmatrix} = \begin{pmatrix} 15 \\ -5 \end{pmatrix} + \begin{pmatrix} -4 \\ 8 \end{pmatrix} = \begin{pmatrix} 11 \\ 3 \end{pmatrix}$ So, the vector that is called $\begin{pmatrix} 5 \\ 2 \end{pmatrix}$ in the B-world is the very same vector that is called $\begin{pmatrix} 11 \\ 3 \end{pmatrix}$ in our familiar standard world. The vector itself is invariant; only its description has changed.

The power of a basis lies in its ability to provide a unique address for every single vector in the space. If you could describe the same vector with two different sets of coordinates using the same basis, your coordinate system would be useless—it would be like having two different addresses for the same house. The uniqueness of representation is the defining, non-negotiable property of a basis. This principle is so fundamental that we can use it to solve problems that seem purely algebraic, by reminding ourselves of the underlying structure. A vector $\mathbf{v}$ might be written in a complicated way, but once you know it's being described by the same basis vectors, the coefficients for each basis vector must be identical.

If different bases are like different languages for describing vectors, how do we translate from one to another? This is not just a mathematical curiosity; it's a practical necessity in fields like computer graphics and engineering. A shape might be defined in a "local" coordinate system that's easy to work with (e.g., centered on the object itself), but to place it in a larger "world," it needs to be translated into a global coordinate system.

The process is always a two-step dance, using the standard basis as a common ground, a lingua franca.

1. ​​Decode:​​ Take the coordinates in the starting basis (say, basis $B$) and use the definition of that basis to find the vector's representation in the standard basis.
2. ​​Encode:​​ Take the resulting standard vector and figure out the recipe—the new coordinates—needed to build it using the target basis (say, basis $C$).

Let's see this in action. A vector $\mathbf{v}$ has coordinates $\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ with respect to basis $B = \{\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}\}$. We want its coordinates in basis $C = \{\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\}$.

​​Step 1 (Decode):​​ First, what is the vector $\mathbf{v}$? $\mathbf{v} = 2 \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} - 1 \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} + 3 \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 2-0+3 \\ 0-1+3 \\ 2-1+0 \end{pmatrix} = \begin{pmatrix} 5 \\ 2 \\ 1 \end{pmatrix}$ This is our vector in the universal standard language.

​​Step 2 (Encode):​​ Now, how do we write $\begin{pmatrix} 5 \\ 2 \\ 1 \end{pmatrix}$ using basis $C$? We need to find scalars $c_1, c_2, c_3$ such that: $c_1 \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + c_2 \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} + c_3 \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 5 \\ 2 \\ 1 \end{pmatrix}$ This leads to a simple system of equations: $c_1+c_2=5$, $c_1-c_2=2$, and $c_3=1$. Solving this gives us $c_1 = \frac{7}{2}$, $c_2 = \frac{3}{2}$, and $c_3=1$. The new coordinates are $\begin{pmatrix} 7/2 \\ 3/2 \\ 1 \end{pmatrix}$. Same vector, different recipe.

This exact same logic applies even when our "vectors" are not arrows but other mathematical objects like polynomials. A basis for polynomials of degree at most 1, like $\{1, t-a\}$, allows us to express any such polynomial in terms of its value at $t=a$ and its slope. This is the first step towards the idea of a Taylor series, a profoundly useful tool in all of science.

So we can choose any basis we like. Are some choices better than others? Emphatically, yes! Imagine navigating a city where the streets cross at all sorts of strange angles. It would be a nightmare. We love city grids where streets are perpendicular because moving east doesn't affect how far north you've gone. The two directions are independent.

This is the physical intuition behind an ​​orthogonal basis​​—a basis where all the vectors are mutually perpendicular (their dot product is zero). Working with an orthogonal basis is a joy because it lets you find each coordinate independently.

In a general basis, finding coordinates means solving a tangled web of simultaneous equations. But if you have an orthogonal basis $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$, the coordinates $c_i$ of a vector $\mathbf{x}$ are found by simply "projecting" $\mathbf{x}$ onto each basis vector: $c_i = \frac{\mathbf{x} \cdot \mathbf{v}_i}{\mathbf{v}_i \cdot \mathbf{v}_i}$ Each coordinate is a simple ratio: "how much of $\mathbf{x}$ is aligned with $\mathbf{v}_i$," corrected for the length of the measuring stick $\mathbf{v}_i$. Notice how the calculation for $c_1$ doesn't involve $\mathbf{v}_2$ or $\mathbf{v}_3$ at all! The components are decoupled. This is a tremendous simplification.

We can take this one step further. What if we also require our basis vectors to have a length of one? This is called an ​​orthonormal basis​​. It's the mathematical equivalent of a perfect grid of meter sticks. In this case, the denominator in our formula, $\mathbf{v}_i \cdot \mathbf{v}_i = \|\mathbf{v}_i\|^2$, is just $1$. The formula for coordinates becomes breathtakingly simple: $c_i = \mathbf{x} \cdot \mathbf{v}_i$ This is why orthonormal bases are the gold standard in physics and engineering. In signal processing, they allow us to decompose a complex signal into a sum of simple, pure frequencies. In quantum mechanics, the state of a particle is a vector, and we find the probability of observing a certain outcome by calculating its coordinates with respect to an orthonormal basis of possible states.

A beautiful consequence of this structure is that if a vector happens to lie in a subspace spanned by a subset of the basis vectors (say, a vector in the xy-plane within a 3D space), its coordinates corresponding to any basis vector outside that subspace will naturally be zero. This is the mathematical formalization of the obvious: if your car is driving on a flat plane, its "altitude" coordinate must be zero.

Changing our basis changes the coordinate numbers. But it doesn't change the vector itself. This means that intrinsic properties of the vector, like its length, must remain the same no matter what coordinate system we use to describe them. This idea of ​​invariance​​ is one of the deepest and most powerful in all of physics.

Let's take a vector $\mathbf{v}$ in 3D space. Its length squared in the standard basis is $\|\mathbf{v}\|^2 = v_1^2 + v_2^2 + v_3^2$. This is just the Pythagorean theorem. Now, what happens if we measure this vector in a new, shiny ​​orthonormal​​ basis and get new coordinates $(c_1, c_2, c_3)$? Remarkably, we find that: $c_1^2 + c_2^2 + c_3^2 = v_1^2 + v_2^2 + v_3^2 = \|\mathbf{v}\|^2$ This is a generalization of the Pythagorean theorem, often called ​​Parseval's Identity​​. It's a statement of profound beauty: the length of a vector is a true, invariant property. Our choice of (orthonormal) coordinates can't change it. Different observers, using different rulers oriented in different ways, will all agree on the vector's length. This is the geometric heart of the matter.

The relationship between a vector and its coordinates is a deep one. What if we sought out a special vector—a vector whose components in the standard basis are exactly the same as its coordinates in some other basis? This is a strange request, like looking for an object that looks identical to its own blueprint. Solving such a problem forces us to confront the meaning of coordinates head-on. It's a bridge between the abstract description and the concrete object, and finding such a vector reveals a special direction in the space that is "stable" under that particular change of basis—a precursor to the all-important idea of eigenvectors.

In the end, the study of coordinates and bases is the study of perspective. By understanding how to choose our point of view wisely, we can untangle complex problems, revealing the simple, beautiful, and invariant truths that lie beneath the surface.

Now that we have grappled with the machinery of changing our perspective—of describing vectors using different sets of basis vectors—it's time for the fun part. Why did we go to all this trouble? The answer, you see, is that the world rarely comes in a neat, pre-packaged Cartesian box. Nature has its own preferred directions, its own symmetries, its own peculiar geometries. The art of the scientist, the engineer, and the mathematician often lies not in forcing a problem into a preconceived framework, but in finding the right framework—the right basis—that makes the problem's inner beauty and simplicity shine through. Choosing a basis is choosing a language, and the right language can turn a tangled mess into an elegant statement.

Let's begin in the physical world. Imagine you are a physicist studying a new crystalline material. You apply an electric field and measure the material's response. You quickly discover that the material is anisotropic—it reacts differently depending on the direction of the applied field. It has a "grain," an internal structure with principal axes along which its physical properties are most simply described. It makes all the sense in the world to describe the input field using these natural axes as your basis. However, your detectors are bolted to your lab bench, which is aligned to the standard north-south, east-west laboratory coordinates. You are in a classic "change of basis" situation: you must 'speak' to the crystal in its own language (the basis of its principal axes) but 'listen' for its reply in your language (the standard lab basis). The matrix that connects these two descriptions is the key to your computational model, a concrete tool born from an abstract idea.

This idea of a "natural" basis is everywhere in the solid-state world. Consider the regular, repeating structure of a salt crystal. The atoms form a beautiful, three-dimensional pattern called a lattice. To specify the location of an atom, would you use its coordinates in millimeters from the corner of the room? Of course not! You would use the lattice's own repeating vectors as your basis. Any position in the crystal can then be described as stepping a certain number of lattice vectors and then adding a small displacement within a single "unit cell". The coordinates in this basis are called fractional coordinates, and they are the native language of crystallography. What's more, there isn't just one way to choose these basis vectors for a given lattice; multiple "primitive cells" exist. Changing from one to another is, you guessed it, just another change of basis.

So far, our notion of "length" and "angle" has been the familiar one from Euclidean geometry. But what if the very definition of distance changes? Imagine a space where moving "north" is twice as difficult as moving "east." We could define an inner product that reflects this, perhaps like $\langle u, v \rangle = 2u_1v_1 + u_2v_2$. In this world, the standard basis vectors $\mathbf{e}_1=(1,0)$ and $\mathbf{e}_2=(0,1)$ are no longer "orthogonal" in the sense of this new inner product! The very concept of orthogonality is relative. The Gram-Schmidt process we learned can be used to find a new basis that is orthonormal with respect to this new rule for measuring length and angle. Finding a vector's coordinates in this new basis simplifies calculations enormously, because in this basis, the "effort" of moving along a vector is once again just the sum of the squares of its coordinates. This is a profound leap: we are tailoring not just our basis, but our very definition of geometry to suit the problem.

As we venture into the realms of quantum mechanics and general relativity, the vector spaces become more abstract, but the importance of choosing the right coordinates only grows.

In the quantum world, the state of a system is a vector in a complex vector space. Physical observables, like momentum or energy, are represented by operators, which can be thought of as matrices. Consider the spin of an electron, a purely quantum property. The "space" of $2 \times 2$ Hermitian matrices, which represent the observables for such a system, has a wonderfully convenient basis: the identity matrix and the three Pauli matrices, $\sigma_1, \sigma_2, \sigma_3$. These matrices are the building blocks of spin. Expressing a given observable-matrix in this basis is like asking: "How much 'spin-x' character, how much 'spin-y' character, and how much 'spin-z' character does this physical quantity have?" The coordinates are not just numbers; they are a physical decomposition into fundamental components. This is the language of quantum computing, where the Pauli matrices form the basis for operations on a qubit.

Then there is Einstein's magnificent theory of general relativity. In a curved spacetime, there are no universal, straight-line coordinate axes. All coordinate systems are local, like drawing a small grid on the curved surface of an orange. The geometry of this curved space is captured by a single object: the metric tensor, $g_{ij}$. This tensor is the star of the show. It tells you everything about the local geometry. And what is it? It's simply the matrix of inner products of your local basis vectors: $g_{ij} = g(\partial_i, \partial_j)$. The formula for the squared length of a vector $v$ in this coordinate system is the generalization of Pythagoras's theorem to curved space: $|v|^2 = g_{ij} v^i v^j$. This is the heart of Riemannian geometry.

This leads to a subtle but crucial point. In a non-Euclidean world, we must distinguish between two types of vectors. The metric tensor $g_{ij}$ lets us do this by defining a "dual basis". In a skewed, non-orthogonal coordinate system, the dual basis vectors provide a reciprocal framework necessary for correctly measuring things like components of forces or gradients of fields. And in a moment of pure mathematical elegance, it turns out that the matrix that converts original basis coordinates into dual basis coordinates is simply the inverse of the metric tensor matrix! This deep connection between geometry (the metric) and algebra (the matrix inverse) is one of the most beautiful revelations in physics.

The power of coordinates extends far beyond vectors-as-arrows. Mathematicians realized that anything that obeys the rules of vector addition and scalar multiplication can be treated as a vector.

Consider the space of all polynomials of degree at most 2. A polynomial like $a+bt+ct^2$ can be thought of as a vector whose coordinates in the standard basis $\{1, t, t^2\}$ are simply $(a, b, c)$. We can define linear transformations on these polynomial spaces, just as we do for geometric vectors. And just as before, we can represent these transformations as matrices, but the matrix will depend on our choice of basis. Changing from one basis of polynomials to another—say, from $\{1-t, t-t^2, 1+t^2\}$—requires the same change-of-basis machinery. This is not just a game. In computer graphics, curves are often defined using a basis of Bernstein polynomials; changing the coordinates of the "control points" elegantly reshapes the curve.

The same principle applies to spaces of matrices themselves, or to more complex objects like bilinear forms. The unifying theme is that by representing abstract objects as coordinate vectors, we can use the powerful and concrete tools of matrix algebra to analyze and manipulate them.

Perhaps the ultimate demonstration of this problem-solving paradigm comes from the abstract world of number theory. A famous result called Minkowski's theorem relates the volume of a convex shape to whether it must contain a point from a given lattice. A problem involving a skewed lattice and a complicated shape can be fiendishly difficult. However, by performing a clever change of coordinates—a transformation defined using the dual of the original lattice—the problem can be magically transformed. In the new coordinate system, the skewed lattice becomes the simple integer grid $\mathbb{Z}^n$, and the question becomes much simpler to answer. This is the pinnacle of the art: finding the one perspective, the one special basis, that makes the complex simple.

From probing the heart of a crystal to navigating the cosmos, from drawing a smooth curve on a screen to proving deep theorems about numbers, the concept of a coordinate system is our primary tool for imposing order on the world. It is a testament to the power of a simple idea: that our understanding depends fundamentally on our point of view.