Linear Algebra: Matrix Representation of Transformations

Representing transformations with matrices is the core computational engine of linear algebra, turning abstract operations into concrete, calculable arrays of numbers. This translation is fundamental across engineering disciplines, from simulating physical forces in mechanical systems to manipulating computer graphics and solving complex circuit equations. However, the matrix you get isn't absolute; it's a description relative to your chosen coordinate system, and understanding this relationship is key to wielding linear algebra with power and precision.

Constructing the Matrix of a Linear Transformation

A linear transformation $T:V\to W$ is a function between vector spaces that preserves vector addition and scalar multiplication. To encode this function as a matrix, you must first choose an ordered basis for the input space $V$ and an ordered basis for the output space $W$.

The process is algorithmic and powerful. Given a basis $\mathcal{B}=\{{\mathbf{v}}_1,{\mathbf{v}}_2,...,{\mathbf{v}}_n\}$ for $V$ and a basis $\mathcal{C}$ for $W$, the matrix $[T{]}_{\mathcal{C}\leftarrow \mathcal{B}}$ is constructed column by column:

1. Apply the transformation $T$ to the first basis vector of $\mathcal{B}$: $T({\mathbf{v}}_1)$.
2. Express the output $T({\mathbf{v}}_1)$ as a linear combination of the output basis $\mathcal{C}$.
3. The coefficients (or coordinates) of this combination become the first column of the matrix.
4. Repeat for each basis vector in $\mathcal{B}$.

For example, consider the linear transformation $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ that rotates vectors counterclockwise by $90^{\circ }$. Using the standard basis $\mathcal{E}=\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$ for both input and output:

• $T({\mathbf{e}}_1)=T(1,0)=(0,1)$. In the standard basis, $(0,1)=0\cdot {\mathbf{e}}_1+1\cdot {\mathbf{e}}_2$. First column: $\left(\begin{array}{c}0\\ 1\end{array}\right)$.
• $T({\mathbf{e}}_2)=T(0,1)=(-1,0)$. This is $(-1)\cdot {\mathbf{e}}_1+0\cdot {\mathbf{e}}_2$. Second column: $\left(\begin{array}{c}-1\\ 0\end{array}\right)$.

Thus, the matrix representation is $[T{]}_{\mathcal{E}}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$. The beauty of this representation is that for any input vector $\mathbf{x}$ with coordinate vector $[\mathbf{x}{]}_{\mathcal{E}}$, you can compute the coordinates of the output by simple matrix multiplication: $[T(\mathbf{x}){]}_{\mathcal{E}}=[T{]}_{\mathcal{E}}[\mathbf{x}{]}_{\mathcal{E}}$.

The Effect of Basis Choice on Matrix Form

The same linear transformation can look completely different depending on the bases chosen. This is because the matrix doesn't describe the transformation in a vacuum; it describes how the transformation acts on coordinates. Changing the basis changes the coordinate system, and thus changes the map of coordinates.

Consider a simple projection in ${\mathbb{R}}^2$ onto the line $y=x$. In the standard basis, its matrix is $\frac{1}{2}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$. Now, choose a basis adapted to the transformation: $\mathcal{B}=\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$ where ${\mathbf{u}}_1=(1,1)$ lies on the line of projection and ${\mathbf{u}}_2=(1,-1)$ is perpendicular to it.

• $T({\mathbf{u}}_1)={\mathbf{u}}_1=1\cdot {\mathbf{u}}_1+0\cdot {\mathbf{u}}_2$.
• $T({\mathbf{u}}_2)=0=0\cdot {\mathbf{u}}_1+0\cdot {\mathbf{u}}_2$.

The matrix relative to $\mathcal{B}$ becomes $[T{]}_{\mathcal{B}}=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$, a diagonal matrix. The transformation hasn't changed, but its matrix representation is vastly simpler. This illustrates a central engineering pursuit: finding bases that reveal the simplest, most informative structure of a transformation.

Similar Matrices: Representing the Same Transformation

When you represent a transformation from a vector space to itself ($T:V\to V$), you typically use the same basis $\mathcal{B}$ for both input and output. The matrix is denoted $[T{]}_{\mathcal{B}}$. If you then change to a new basis ${\mathcal{B}}^{\prime }$, how are the two matrices $[T{]}_{\mathcal{B}}$ and $[T{]}_{{\mathcal{B}}^{\prime }}$ related?

They are similar matrices. Two $n\times n$ matrices $A$ and $B$ are similar if there exists an invertible change-of-basis matrix $P$ such that $B=P^{-1}AP$. Here, the columns of $P$ are the coordinates of the new basis ${\mathcal{B}}^{\prime }$ vectors expressed in the old basis $\mathcal{B}$. The equation $B=P^{-1}AP$ is the algebraic manifestation of changing your perspective. Similar matrices represent the same linear operator, just in different languages (bases). They share fundamental properties like determinant, trace, rank, eigenvalues, and characteristic polynomial, which is why these are called similarity invariants.

Finding Bases that Simplify Representation

The ultimate goal is computational efficiency and insight. The simplest useful form for a matrix is a diagonal matrix, where application of the transformation is just scalar multiplication on each coordinate. This leads to the concepts of eigenvectors and diagonalization.

A non-zero vector $\mathbf{v}$ is an eigenvector of a linear operator $T$ if $T(\mathbf{v})=\lambda \mathbf{v}$ for some scalar $\lambda$, the eigenvalue. If you can form a basis $\mathcal{B}$ for $V$ consisting entirely of eigenvectors of $T$, then $[T{]}_{\mathcal{B}}$ is a diagonal matrix with the corresponding eigenvalues on the diagonal. In this basis, applying $T$ to a vector is trivial: you just multiply each coordinate by its corresponding eigenvalue. This makes computing powers of the transformation ($T^k$) extraordinarily efficient.

Not all transformations are diagonalizable. In such cases, you aim for the next best form, like the Jordan canonical form, which is nearly diagonal and still simplifies many computations. In engineering, diagonalizing a matrix representing a stress tensor reveals the principal axes of stress. Simplifying the matrix representation of a system's dynamics matrix can decouple complex differential equations into independent, solvable ones.

Common Pitfalls

1. Forgetting that a matrix is basis-dependent. A common error is to treat a given matrix as the "true" representation of a transformation. Always remember the implicit "with respect to the standard basis" clause. Stating "the matrix of the transformation is..." is incomplete without specifying the bases.

• Correction: Always explicitly note the bases, e.g., $[T{]}_{\mathcal{C}\leftarrow \mathcal{B}}$, or clarify the context (e.g., "relative to the standard basis").

1. Confusing the change-of-basis formula. When converting coordinates of a vector from basis $\mathcal{B}$ to ${\mathcal{B}}^{\prime }$, you use the change-of-basis matrix $P$: $[\mathbf{x}{]}_{{\mathcal{B}}^{\prime }}=P^{-1}[\mathbf{x}{]}_{\mathcal{B}}$. When converting the matrix of a transformation, you conjugate by $P$: $[T{]}_{{\mathcal{B}}^{\prime }}=P^{-1}[T{]}_{\mathcal{B}}P$. Mixing these up reverses the direction of the change.

• Correction: Remember the mnemonic: For a vector, new coordinates = $P^{-1}$ (old coordinates). For an operator, new matrix = $P^{-1}$ (old matrix) $P$.

1. Assuming diagonalizability. It is tempting to always look for a diagonal matrix. However, a matrix is diagonalizable only if there exists a basis of eigenvectors. A matrix with insufficient eigenvectors (geometric multiplicity < algebraic multiplicity for some eigenvalue) cannot be diagonalized.

• Correction: Before attempting to diagonalize, check that for each eigenvalue, the dimension of its eigenspace equals its multiplicity as a root of the characteristic polynomial.

Summary

• A matrix is merely a representation of a linear transformation, contingent on specific choices of ordered bases for the input and output spaces.
• Changing the basis changes the matrix representation. Matrices representing the same linear operator in different bases are similar, related by $B=P^{-1}AP$, and share key invariants like eigenvalues and determinant.
• The choice of basis is a powerful tool for computational efficiency. Finding a basis of eigenvectors, when possible, yields a diagonal matrix representation, making analysis and iteration of the transformation vastly simpler.
• The core workflow involves moving between the abstract transformation, its concrete matrix representation in a given basis, and the coordinate vectors of points, linked by the fundamental equation $[T(\mathbf{x})