70. Projection

The projection of a vector , onto a line , denoted as , is a vector that lies on the line , such that the difference between and its projection, , is orthogonal to

can be seen as the “shadow”cast by onto when light shines perpendicularly to .

Equivalently

Where:

Example
  1. Define the line as all vectors of the form , where is a scalar:
  1. Compute the projection using

Projection as a Transformation

As a matrix vector product

If is a unit vector:

Then

If we redefine our line as all the scalar multiples of out unit vector :

Simplifies to:

Projection as a Linear Transformation

Let be a unit vector:

The projection matrix is:

Expands to:

For any , the projection of onto the line spanned by is:

Example

Consider a vector in :

1. Construct the Unit Vector

This unit vector defines the line , which consists of all the scalar multiples og :

2. Derive the Projection Matrix

The projection of any vector onto the line is given by:

Where is the projection matrix. To construct we use the formula:

3. Applying the Projection

To project any vector onto , we multiply by the matrix :

  1. Additivity of Projections (Linearity with respect to addition)
  1. Homogeneity of Projections (Linearity with respect to scalar multiplication)

General Properties of

  1. Idempotence
  1. Symmetry
  1. Rank

Because projects onto a one-dimensional subspace spanned by