Changing Worlds: A Storyteller's Guide to Change of Basis

Once upon a time, in the vast, geometric kingdom of Vector Space, everything was described in relation to two grand, perpendicular roads: the x-axis and the y-axis. Every location, every journey, was meticulously plotted using these roads as a reference. This was the 'Standard Basis', the common tongue that everyone understood. But what if you lived on a long, slanted mountain ridge? Wouldn't it be more convenient to describe your world in terms of 'along the ridge' and 'up/down the ridge'? This is the very heart of changing basis. It’s not about changing the world, but changing your perspective of it to make things simpler, more intuitive, and computationally elegant. Prepare yourself, for we are about to embark on a journey to master this new perspective. We will travel from the foundational concepts to the magical realm of eigenvectors, and you will emerge a master of changing worlds.

Part 1: The Language of Vectors - Understanding Basis

Before we can change our language, we must first understand what one is. In linear algebra, our 'language' is a basis. A basis for a vector space is a set of vectors that can be used to 'build' any other vector in that space through scaling and addition.

For the 2D plane (denoted $${\mathbb{R}}^2$$), the most common language is the standard basis, which consists of two vectors:

$$ \vec{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \quad \text{and} \quad \vec{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} $$

When we write a vector $$ \vec{v} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} $$, what we're implicitly saying is:

$$ \vec{v} = 3 \cdot \vec{i} + 2 \cdot \vec{j} = 3 \begin{pmatrix} 1 \\ 0 \end{pmatrix} + 2 \begin{pmatrix} 0 \\ 1 \end{pmatrix} $$

The numbers (3, 2) are the coordinates of our vector in the standard basis. They are the instructions for how to build our vector using the standard language.

Part 2: Speaking a New Language - The Change of Basis

Now, let's introduce a new language, a new basis $$ B $$. Let's say our new basis vectors, described in the old language, are:

$$ \vec{b_1} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \quad \text{and} \quad \vec{b_2} = \begin{pmatrix} -1 \\ 1 \end{pmatrix} $$

These vectors point diagonally. For someone living on a grid tilted at 45 degrees, this basis might be far more natural. Our vector $$ \vec{v} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} $$ still exists in the same place, but how do we describe it using $$ \vec{b_1} $$ and $$ \vec{b_2} $$? We need to find new coordinates $$ c_1 $$ and $$ c_2 $$ such that:

$$ \vec{v} = c_1 \vec{b_1} + c_2 \vec{b_2} $$

$$ \begin{pmatrix} 3 \\ 2 \end{pmatrix} = c_1 \begin{pmatrix} 1 \\ 1 \end{pmatrix} + c_2 \begin{pmatrix} -1 \\ 1 \end{pmatrix} $$

This is a system of linear equations. The solution is $$ c_1 = 2.5 $$ and $$ c_2 = -0.5 $$. So, in our new basis $$ B $$, the vector $$ \vec{v} $$ has coordinates $$ (2.5, -0.5) $$. We write this as $$ [\vec{v}]_B = \begin{pmatrix} 2.5 \\ -0.5 \end{pmatrix} $$. Same vector, different description.

The Universal Translator: The Change of Basis Matrix

Solving a system of equations every time is tedious. We need a 'universal translator'. This is the change of basis matrix.

Let's define a matrix $$ P $$ whose columns are our new basis vectors:

$$ P = [ \vec{b_1} | \vec{b_2} ] = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} $$

This matrix $$ P $$ translates from the new basis to the standard basis. If you have coordinates in basis $$B$$, you can find the standard coordinates ($$S$$) like this:

$$ [\vec{v}]_S = P \cdot [\vec{v}]_B $$

Let's check: $$ \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 2.5 \\ -0.5 \end{pmatrix} = \begin{pmatrix} 2.5 + 0.5 \\ 2.5 - 0.5 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} $$. It works!

But what we usually want is the reverse: to go from standard coordinates to our new basis. For that, we need the inverse matrix, $$ P^{-1} $$.

$$ [\vec{v}]_B = P^{-1} \cdot [\vec{v}]_S $$

Part 3: A New Point of View - Transforming Transformations

Here's where the real power emerges. Matrices don't just hold numbers; they represent linear transformations—actions like rotations, reflections, and shears. But the matrix that represents a transformation depends on the basis you're using!

Imagine a transformation $$ T $$ represented by matrix $$ A $$ in the standard basis. If we apply it to our vector $$ \vec{v} $$, the result is $$ A\vec{v} $$. Now, what if we want a matrix $$ D $$ that performs the exact same transformation, but works entirely in our new basis $$ B $$? It should take a vector's $$ B $$-coordinates and output the transformed vector's $$ B $$-coordinates.

This leads to one of the most important formulas in linear algebra, the similarity transformation:

$$ D = P^{-1}AP $$

The matrix $$ D $$ is the transformation $$ A $$ as seen from the perspective of basis $$ B $$. The matrices $$ A $$ and $$ D $$ are 'similar'—they represent the exact same geometric operation, just described in different languages.

Part 4: The Payoff - The Perfect Perspective of Eigen Decomposition

Why go through all this trouble? Because we can choose a basis that is perfectly suited for a given transformation, making its matrix representation incredibly simple. This perfect basis is the eigenbasis.

What are Eigenvectors?

For a given transformation matrix $$ A $$, an eigenvector is a special, non-zero vector that, when the transformation is applied, does not change its direction. It only gets stretched or shrunk by a certain factor. This factor is its corresponding eigenvalue ($$\lambda$$).

$$ A\vec{v} = \lambda \vec{v} $$

Eigenvectors are the 'axes' of a transformation. They reveal the fundamental directions along which the transformation acts purely as a scaling.

Let's use our change of basis machinery. We'll choose our new basis $$ B $$ to be the set of eigenvectors of $$ A $$. Our change of basis matrix $$ P $$ will have these eigenvectors as its columns.

$$ P = [ \vec{v_1} | \vec{v_2} | \dots | \vec{v_n} ] $$

What will the new matrix $$ D = P^{-1}AP $$ look like? It will be a beautifully simple diagonal matrix, with the eigenvalues on the diagonal!

$$ D = \begin{pmatrix} \lambda_1 & 0 & \dots & 0 \\ 0 & \lambda_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \lambda_n \end{pmatrix} $$

This is called diagonalization, and it is the ultimate 'convenient perspective'.

Why Eigen Decomposition is So Convenient

1. Understanding Geometry: The equation $$ A = PDP^{-1} $$ (just rearranging the similarity transform) decomposes the complex action of $$ A $$ into three simple steps: change to the eigenbasis ($$P^{-1}$$), perform a simple scaling along the eigen-axes ($$D$$), and change back to the standard basis ($$P$$). This reveals the true geometric nature of the transformation.
2. Computational Power: Calculating powers of $$ A $$ is difficult. But powers of $$ D $$ are easy—just raise the diagonal elements to that power. Then, calculating $$ A^k $$ becomes much simpler: $$ A^k = (PDP^{-1})^k = (PDP^{-1})(PDP^{-1})\dots(PDP^{-1}) = PD^kP^{-1} $$ This is crucial in fields like physics, computer graphics, and machine learning for modeling systems that evolve over time.

Example: Diagonalizing a Shear Transformation

Consider the shear matrix $$ A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix} $$. Its eigenvectors and eigenvalues are:

• $$ \lambda_1 = 1 $$ with eigenvector $$ \vec{v_1} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} $$
• $$ \lambda_2 = 3 $$ with eigenvector $$ \vec{v_2} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} $$

Our eigenbasis is $$B = \{ \vec{v_1}, \vec{v_2} \}$$. The change-of-basis matrix $$P$$ and its inverse $$P^{-1}$$ are:

$$ P = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \quad \text{and} \quad P^{-1} = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} $$

Now, let's find the transformation matrix $$ D $$ in the eigenbasis:

$$ D = P^{-1}AP = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} $$

$$ D = \begin{pmatrix} 1 & -1 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} $$

As predicted! In the world defined by its eigenvectors, the complicated shear transformation is just a simple scaling: it scales by 1 in the $$ \vec{v_1} $$ direction and by 3 in the $$ \vec{v_2} $$ direction.

Conclusion: Mastering Perspectives

Our journey is complete. We began by understanding a basis as a language. We learned how to translate vectors between languages ($$[\vec{v}]_B = P^{-1}[\vec{v}]_S$$) and, more powerfully, how to translate entire transformations ($$D = P^{-1}AP$$). Finally, we discovered the 'perfect language'—the eigenbasis—which simplifies a transformation to its fundamental scaling actions, a process called diagonalization.

Changing basis is far more than an algebraic trick. It's a fundamental problem-solving strategy that echoes across science and engineering. It teaches us that when a problem seems impossibly complex, it's often because we're looking at it from the wrong perspective. By finding the right 'basis', the right frame of reference, the solution can become not just manageable, but beautifully simple. You are now equipped with one of the most powerful tools in mathematics—the ability to change your world to better understand it.