The Matrix of a Linear Transformation

The matrix of a linear transformation is simply a record of where the transformation sends each axis direction. All we're really saying is: if you know where the unit arrows along the axes land, you know where everything lands, because every vector is built out of those arrows. That's good because it turns any geometric move, like "rotate by 30°" or "reflect across this line", into a matrix you can actually compute with. The formula above has just two kinds of pieces: $\vec e_1, \dots, \vec e_n$ (the standard basis vectors, the unit arrows along each axis, like $\binom10$ and $\binom01$ in the plane) and $T(\vec e_1), \dots, T(\vec e_n)$ (where each of those arrows ends up, which become the columns, in order). It's one column per axis; bear with it.

Why it works. This builds right on the last lesson's point that knowing a few outputs tells you all of them. Any vector can be written as $\vec x = x_1\vec e_1 + \cdots + x_n\vec e_n$ (for example, $\binom35 = 3\vec e_1 + 5\vec e_2$), so by linearity $T(\vec x) = x_1T(\vec e_1) + \cdots + x_nT(\vec e_n)$, which is exactly $A\vec x$ with those columns. Notice that last step is the column picture of $A\vec x$ from earlier in the unit: the weights $x_1, \dots, x_n$ go on the columns. So every linear transformation from $\mathbb R^n$ to $\mathbb R^m$ is a matrix in disguise.

Building matrices geometrically: ask where the unit vectors land. A 90° rotation sends $\vec e_1 \to \vec e_2$ and $\vec e_2 \to -\vec e_1$, so its matrix is $\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$. A horizontal shear keeps $\vec e_1$ and tilts $\vec e_2$ to $\binom k1$. You don't need a formula for any of these; a quick sketch of where the two arrows go is enough. The same thinking gives the rotation matrix $R_\theta$ above: turning $\vec e_1$ by $\theta$ lands it at $(\cos\theta, \sin\theta)$, and turning $\vec e_2$ lands it at $(-\sin\theta, \cos\theta)$.

A common mistake. It's tempting to write $T(\vec e_1)$ as the first row of the matrix, because we read a matrix row by row. It doesn't go there: the images go in as columns. Getting this backward gives you the transpose, which for a rotation is the rotation the other way. A quick check: $A\vec e_1$ always picks out the first column of $A$, so that's where $T(\vec e_1)$ has to be.

One-to-one and onto. $T$ is one-to-one when the columns are independent (pivot in every column), and onto when the columns span the codomain (pivot in every row). Both come straight from earlier lessons. One-to-one means no two different inputs land on the same output. If two inputs $\vec u \ne \vec v$ did, then $A(\vec u - \vec v) = \vec 0$ with $\vec u - \vec v \ne \vec 0$, and that's exactly a dependence among the columns. Onto means every $\vec b$ is an output, which is the "solution for every $\vec b$" question from the $A\vec x = \vec b$ lesson.

There's an animation further down where you slide a shear and watch its second column move.

The robot arm lab builds each joint's rotation matrix this way, and camera projection matrices in computer vision are found by asking where the axes go.