Question

Let \(L\) be the line through the origin in \(\mathbb{R}^{2}\) with direction vector \(\mathbf{d}=\left[\begin{array}{l}a \\ b\end{array}\right] \neq 0\) a. If \(P_{L}\) denotes projection on \(L\), show that \(P_{L}\) has matrix \(\frac{1}{a^{2}+b^{2}}\left[\begin{array}{cc}a^{2} & a b \\ a b & b^{2}\end{array}\right]\) b. If \(Q_{L}\) denotes reflection in \(L,\) show that \(Q_{L}\) has ma\(\operatorname{trix} \frac{1}{a^{2}+b^{2}}\left[\begin{array}{cc}a^{2}-b^{2} & 2 a b \\ 2 a b & b^{2}-a^{2}\end{array}\right]\)

Short answer

The projection matrix is \(\frac{1}{a^2+b^2}\begin{bmatrix} a^2 & ab \\ ab & b^2 \end{bmatrix}\) and the reflection matrix is \(\frac{1}{a^2+b^2}\begin{bmatrix} a^2-b^2 & 2ab \\ 2ab & b^2-a^2 \end{bmatrix}\).

Step-by-step solution

Express a Point on the Line

A line through the origin in the direction \( \mathbf{d} = \begin{bmatrix} a \ b \end{bmatrix} \) can be expressed as \( t \mathbf{d} = \begin{bmatrix} ta \ tb \end{bmatrix} \), where \( t \) is a scalar.

Define the Projection Formula

The projection of a vector \( \mathbf{v} = \begin{bmatrix} x \ y \end{bmatrix} \) onto \( \mathbf{d} \) is given by the formula: \[ P_{L}(\mathbf{v}) = \frac{\mathbf{v} \cdot \mathbf{d}}{\mathbf{d} \cdot \mathbf{d}} \mathbf{d} \]

Compute Dot Products

Compute the dot products:\[ \mathbf{v} \cdot \mathbf{d} = ax + by \] \[ \mathbf{d} \cdot \mathbf{d} = a^2 + b^2 \]

Substitute Into Projection Formula

Substitute the dot products into the projection formula:\[ P_{L}(\mathbf{v}) = \frac{ax + by}{a^2 + b^2} \begin{bmatrix} a \ b \end{bmatrix} = \begin{bmatrix} \frac{a(ax + by)}{a^2 + b^2} \ \frac{b(ax + by)}{a^2 + b^2} \end{bmatrix} \]

Write the Projection Matrix

Express \( P_{L}(\mathbf{v}) \) in matrix form:\[ P_{L} = \frac{1}{a^2 + b^2} \begin{bmatrix} a^2 & ab \ ab & b^2 \end{bmatrix} \]

Apply Reflection Formula

The reflection of \( \mathbf{v} \) across the line \( L \) can be expressed as:\[ Q_{L}(\mathbf{v}) = 2P_{L}(\mathbf{v}) - \mathbf{v} \]

Substitute Projection Matrix

Substitute \( P_{L}(\mathbf{v}) \) into the reflection formula:\[ Q_{L} = 2 \times \frac{1}{a^2 + b^2} \begin{bmatrix} a^2 & ab \ ab & b^2 \end{bmatrix} - \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]

Simplify Reflection Matrix

Simplify the expression to obtain:\[ Q_{L} = \frac{1}{a^2 + b^2} \begin{bmatrix} a^2 - b^2 & 2ab \ 2ab & b^2 - a^2 \end{bmatrix} \]

Key concepts

reflection matrix

A reflection matrix in linear algebra allows us to reflect vectors across a specified line. For a reflection to occur in the line through the origin with direction vector \( \mathbf{d} = \begin{bmatrix} a \ b \end{bmatrix} eq 0 \), the reflection matrix is constructed such that the transformed vector \( \mathbf{v} \) is equidistant on the opposite side of the line compared to its original position.

Reflection is a linear transformation that flips each point over the given line. To find the reflection matrix \( Q_{L} \), we start by considering the projection matrix \( P_{L} \). The reflection formula combines the original vector and its projection:

• The formula is: \( Q_{L}(\mathbf{v}) = 2P_{L}(\mathbf{v}) - \mathbf{v} \).
• This essentially doubles the effect of the projection and subtracts the original vector to achieve reflection.

The calculated reflection matrix is:\[Q_{L} = \frac{1}{a^2 + b^2} \begin{bmatrix} a^2 - b^2 & 2ab \ 2ab & b^2 - a^2 \end{bmatrix}.\]
This matrix efficiently transforms any given vector \( \mathbf{v} \) uniformly across the line \( L \).

linear algebra

Linear algebra is the branch of mathematics dealing with vector spaces and linear mappings between these spaces. It is fundamental for understanding geometry in high-dimensional spaces, including operations involving matrices and vectors.

In linear algebra, matrices are used to represent linear transformations like rotations, reflections, and projections. Understanding matrices enables us to apply geometric operations to vectors collectively. Here's how:

• Vectors: Vectors represent points in space, direction, and magnitude.
• Matrices: They serve as operators that can transform a set of vectors in space.

Linear algebra provides computational tools necessary for working with these matrices and vectors, enabling calculations of things such as projections (scatter a vector across a line) and reflections. Becoming comfortable with these concepts opens up a wide range of possibilities for solving geometrical and practical problems in various fields.

vector projection

Vector projection is a powerful concept in linear algebra that allows us to project one vector onto another. This provides us with a new vector that lies on the line defined by the direction vector.

When projecting vector \( \mathbf{v} = \begin{bmatrix} x \ y \end{bmatrix} \) onto \( \mathbf{d} = \begin{bmatrix} a \ b \end{bmatrix} \), we use the formula:\[P_{L}(\mathbf{v}) = \frac{\mathbf{v} \cdot \mathbf{d}}{\mathbf{d} \cdot \mathbf{d}} \mathbf{d}\]
The steps involve:

• Calculating dot products: \( \mathbf{v} \cdot \mathbf{d} = ax + by \), and \( \mathbf{d} \cdot \mathbf{d} = a^2 + b^2 \).
• Substituting these into the projection formula to get: \( \begin{bmatrix} \frac{a(ax + by)}{a^2 + b^2} \ \frac{b(ax + by)}{a^2 + b^2} \end{bmatrix} \).

This yields the matrix form of the projection operator:\[P_{L} = \frac{1}{a^2 + b^2} \begin{bmatrix} a^2 & ab \ ab & b^2 \end{bmatrix}\]

Understanding vector projections helps not only in solving geometric problems but also in applications ranging from computer graphics to physics simulations.