Coordinate Projections and Reflections in R2

Coordinate Actions in the Plane

A projection onto a coordinate axis keeps the component along that axis and removes the other component. A reflection across an axis changes the sign of the perpendicular component, while reflection across the line \(y=x\) exchanges the two coordinates. These familiar geometric actions are linear transformations, so each can be represented by a \(2\times2\) matrix.

Throughout, vectors in \(\mathbb{R}^2\) are columns. The standard basis vectors are \(\mathbf{e}_1=\begin{bmatrix}1\\0\end{bmatrix}\) and \(\mathbf{e}_2=\begin{bmatrix}0\\1\end{bmatrix}\). A point \((x,y)\) is represented by \(\mathbf{x}=\begin{bmatrix}x\\y\end{bmatrix}\). For a linear transformation \(T:\mathbb{R}^2\to\mathbb{R}^2\), its standard matrix has columns \(T(\mathbf{e}_1)\) and \(T(\mathbf{e}_2)\). Indeed, \(\mathbf{x}=x\mathbf{e}_1+y\mathbf{e}_2\), so linearity gives \(T(\mathbf{x})=xT(\mathbf{e}_1)+yT(\mathbf{e}_2)\), which is exactly multiplication by the matrix with those columns.

Each formula is linear: for example, the coordinates of \(P_x(a\mathbf{u}+b\mathbf{v})\) are \(a\) times the first coordinate of \(\mathbf{u}\) plus \(b\) times the first coordinate of \(\mathbf{v}\), followed by a zero second coordinate. The same coordinate-by-coordinate check applies to the other four formulas. Thus the standard-matrix construction applies to every map in the definition.

Constructing the Matrices from Basis Images

The columns must be the images of the basis vectors in order. For a projection onto the \(x\)-axis, \(\mathbf{e}_1\) is already on the axis and stays fixed, while \(\mathbf{e}_2\) projects to \(\mathbf{0}\). For a projection onto the \(y\)-axis, \(\mathbf{e}_1\) projects to \(\mathbf{0}\) and \(\mathbf{e}_2\) stays fixed.

For the axis reflections, the basis vector lying on the reflection axis stays fixed and the other basis vector changes sign. Reflection across \(y=x\) sends \(\mathbf{e}_1\) to \(\mathbf{e}_2\) and \(\mathbf{e}_2\) to \(\mathbf{e}_1\).

Proof. Multiply each matrix by the column \(\begin{bmatrix}x\\y\end{bmatrix}\). The first row and second row of \(\mathbf{P}_x\) give \(x\) and \(0\), respectively; those of \(\mathbf{P}_y\) give \(0\) and \(y\). The rows of \(\mathbf{R}_x\) give \(x\) and \(-y\), and those of \(\mathbf{R}_y\) give \(-x\) and \(y\). Finally, the rows of \(\mathbf{R}_d\) give \(y\) and \(x\). Each product agrees with the corresponding coordinate definition, for every real \(x,y\). \(\square\)

Worked Applications

What Happens When the Transformation Is Repeated?

A projection leaves a vector on its target axis unchanged, so projecting twice has the same effect as projecting once. A reflection sends a vector across its line and a second reflection returns it. These statements can be expressed by matrix equations and verified for every input, not only for a particular drawing.

Proof. Write \(\mathbf{x}=\begin{bmatrix}x\\y\end{bmatrix}\). The first projection gives \(\mathbf{P}_x(\mathbf{x})=\begin{bmatrix}x\\0\end{bmatrix}\); applying \(\mathbf{P}_x\) again gives \(\begin{bmatrix}x\\0\end{bmatrix}\), the same vector. The \(y\)-axis projection gives \(\begin{bmatrix}0\\y\end{bmatrix}\), which is unchanged by another application of \(\mathbf{P}_y\). For the \(x\)-axis reflection, the successive outputs are \(\begin{bmatrix}x\\-y\end{bmatrix}\) and \(\begin{bmatrix}x\\-(-y)\end{bmatrix}=\begin{bmatrix}x\\y\end{bmatrix}\). For the \(y\)-axis reflection, they are \(\begin{bmatrix}-x\\y\end{bmatrix}\) and \(\begin{bmatrix}-(-x)\\y\end{bmatrix}=\begin{bmatrix}x\\y\end{bmatrix}\). For reflection across \(y=x\), they are \(\begin{bmatrix}y\\x\end{bmatrix}\) and \(\begin{bmatrix}x\\y\end{bmatrix}\). Thus each claimed equality holds for every input. \(\square\)

The same facts can be checked by multiplying the matrices. For example, \(\mathbf{P}_x^2=\mathbf{P}_x\), while \(\mathbf{R}_d^2=\mathbf{I}_2\). The first identity describes a projection: once the vector lies on the target axis, another projection does not move it. The second describes a reflection: the transformation is its own inverse, because applying it twice gives the identity transformation. These statements follow from the proved action on every vector.

The fixed vectors also identify the relevant geometric lines. A vector is fixed by a map if its image equals itself. The equation \(\mathbf{P}_x\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x\\y\end{bmatrix}\) becomes \(\begin{bmatrix}x\\0\end{bmatrix}=\begin{bmatrix}x\\y\end{bmatrix}\), which holds exactly when \(y=0\). Thus the fixed vectors of \(\mathbf{P}_x\) are precisely those on the \(x\)-axis. For \(\mathbf{P}_y\), equality requires \(x=0\). For \(\mathbf{R}_x\), equality requires \(-y=y\), hence \(y=0\); for \(\mathbf{R}_y\), it requires \(-x=x\), hence \(x=0\). For \(\mathbf{R}_d\), equality means \(y=x\). In each case, the fixed set is the axis or line named by the transformation.

Common Mistake: Reversing the Matrix Columns

The columns of a standard matrix are images of the basis vectors in the order \(\mathbf{e}_1,\mathbf{e}_2\). A common incorrect matrix for reflection across \(y=x\) is \(\begin{bmatrix}1&0\\0&1\end{bmatrix}\), sometimes chosen because the coordinates of the line itself satisfy \(x=y\). But this matrix fixes every vector: its first column sends \(\mathbf{e}_1\) to \(\mathbf{e}_1\), and its second sends \(\mathbf{e}_2\) to \(\mathbf{e}_2\). It therefore does not exchange coordinates.

The correction is to check the basis images required by the geometry: reflection across \(y=x\) sends \(\mathbf{e}_1\) to \(\mathbf{e}_2\) and \(\mathbf{e}_2\) to \(\mathbf{e}_1\). Place those images as the first and second columns, respectively, to get \(\mathbf{R}_d=\begin{bmatrix}0&1\\1&0\end{bmatrix}\). For instance, the incorrect identity matrix sends \(\begin{bmatrix}2\\-5\end{bmatrix}\) to itself, while the correct matrix sends it to \(\begin{bmatrix}-5\\2\end{bmatrix}\). Checking both basis-vector images is a direct way to catch a column-order error.

Projections and reflections are useful basic models of linear transformations: their coordinate formulas make the geometry explicit, and their standard matrices allow the same actions to be computed by matrix multiplication. When constructing one, first decide where each basis vector goes, then put those two image columns in the correct order. Finally, test a general vector or a specific input to confirm that the matrix matches the intended action.