Matrix Representation of Linear Transformations

1. From the Standard Basis to Any Basis

Let's quickly recall the core idea from last time. For a linear transformation $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$, its matrix A in the standard basis is:

The columns are the images of the standard basis vectors. This works because any vector $\mathbf{x}$ can be written as $x_1{\mathbf{e}}_1+\cdots +x_n{\mathbf{e}}_n$, and its transformation is $T(\mathbf{x})=x_{1}T({\mathbf{e}}_1)+\cdots +x_{n}T({\mathbf{e}}_n)$, which is exactly what matrix-vector multiplication calculates.

Now, let's switch to a new basis $B=\{{\mathbf{b}}_1,{\mathbf{b}}_2,\dots ,{\mathbf{b}}_n\}$. We want to find a matrix, let's call it $[T{]}_B$, that takes the coordinates of a vector in the basis B and gives us the coordinates of the transformed vector, also in the basis B.

The logic is perfectly analogous. The matrix $[T{]}_B$ must be constructed such that its columns tell us where the new basis vectors land, expressed in the new language.

Specifically, the $i$-th column of $[T{]}_B$ is the coordinate vector of $T({\mathbf{b}}_i)$ with respect to the basis $B$. We denote this coordinate vector as $[T({\mathbf{b}}_i){]}_B$.

This gives us a direct, if sometimes tedious, method to build the matrix.

The "Direct" Method in Action

So, how does this work in practice? To find each column $[T({\mathbf{b}}_i){]}_B$, you follow three steps:

1. Take a basis vector from your new basis, ${\mathbf{b}}_i$.
2. Apply the transformation to it to get the output vector $T({\mathbf{b}}_i)$ (this will be in standard coordinates).
3. Find which linear combination of your basis vectors $\{{\mathbf{b}}_1,\dots ,{\mathbf{b}}_n\}$ gives you $T({\mathbf{b}}_i)$. The coefficients of this combination form the coordinate vector $[T({\mathbf{b}}_i){]}_B$, which becomes the $i$-th column of your new matrix.

This process involves solving a system of linear equations for each column.

2. The Intuitive "Change of Perspective" Method

The direct method works, but it feels like a lot of brute-force computation. There's a more elegant and intuitive way to think about this, which frames the problem as a "translation" between different points of view.

Imagine you have a vector described in basis $B$, say $[\mathbf{x}{]}_B$. To apply the transformation $T$, which we know how to do using its standard matrix $A$, we can follow a three-step recipe:

1. Translate to Standard Coordinates: Convert $[\mathbf{x}{]}_B$ from the language of basis $B$ into the standard coordinate language.
2. Apply the Transformation: Use the standard matrix $A$ to transform the vector.
3. Translate back to Basis B: Convert the resulting vector back into the language of basis $B$.

This process is beautifully illustrated in the following video.

Let's formalize this. Let $A$ be the matrix of $T$ in the standard basis. Let $P_B$ be the change-of-basis matrix whose columns are the vectors of our new basis $B$.

• Multiplying by $P_B$ translates a vector's coordinates from basis $B$ to the standard basis: $\mathbf{x}=P_B[\mathbf{x}{]}_B$.
• Multiplying by $P_B^{-1}$ translates from standard coordinates to basis $B$ coordinates: $[\mathbf{x}{]}_B=P_B^{-1}\mathbf{x}$.

Following our three-step recipe, the transformation of $[\mathbf{x}{]}_B$ to $[T(\mathbf{x}){]}_B$ is:

This means the matrix of $T$ in the basis $B$ is the product of these three matrices:

This is a similarity transformation. The matrix $[T{]}_B$ is said to be similar to $A$. It represents the exact same geometric transformation, just described from a different point of view.

Dr. Peyam's video, which you watched earlier, also arrives at this same formula, connecting the procedural method with this more abstract change-of-perspective formula.

3. A Preview: The Power of Eigenbases

Why do we go to all this trouble to change bases? Because we want to find a basis where the transformation's matrix is as simple as possible. The ideal scenario is a diagonal matrix.

Imagine if our basis vectors ${\mathbf{b}}_i$ were special vectors that are only scaled by the transformation, such that $T({\mathbf{b}}_i)={\lambda }_i{\mathbf{b}}_i$. These are called eigenvectors, and the scaling factors ${\lambda }_i$ are eigenvalues.

What would the matrix $[T{]}_B$ look like in this "eigenbasis"?
The first column would be $[T({\mathbf{b}}_1){]}_B=[{\lambda }_1{\mathbf{b}}_1{]}_B$. In the language of $B$, the vector ${\lambda }_1{\mathbf{b}}_1$ is just ${\lambda }_1$ units of the first basis vector and 0 units of the others. So its coordinate vector is simply $\left[\begin{array}{c}{\lambda }_1\\ 0\\ ⋮\\ 0\end{array}\right]$.

Following this logic for all basis vectors, the matrix becomes beautifully simple:

Finding a basis that diagonalizes the matrix of a transformation is one of the most powerful techniques in linear algebra. It's the key to solving systems of differential equations, understanding the core action of quantum operators, and much more.

Conclusion

Today, we've generalized our understanding of matrices. They don't just represent transformations in the standard basis; they can represent a transformation from any point of view.

Key Takeaways:

• The matrix of a linear transformation $T$ depends on the chosen basis $B$.
• The matrix $[T{]}_B$ describes the transformation in the "language" of the basis $B$.
• The $i$-th column of $[T{]}_B$ is the coordinate vector of the transformed basis vector $T({\mathbf{b}}_i)$ relative to $B$, written as $[T({\mathbf{b}}_i){]}_B$.
• This matrix can be found elegantly using the change-of-basis formula: $[T{]}_B=P_B^{-1}A{P}_B$, where $A$ is the standard matrix and $P_B$ is the matrix whose columns are the vectors of basis $B$. This is called a similarity transformation.
• The ultimate goal of changing basis is often to find one where the transformation's matrix becomes simple, ideally diagonal. This occurs when the basis is an eigenbasis.

Preview of the Next Lesson:
We've seen how a matrix describes the overall geometry of a transformation. In the next lesson, we will dissect this geometry by exploring two fundamental subspaces associated with any linear transformation: the null space (or kernel), which consists of all vectors that are squashed into the zero vector, and the column space (or image), which is the set of all possible outputs of the transformation. Understanding these spaces is key to understanding what a transformation does and does not do.