Question

A) In n dimensions, what angle does the vector (1,1,...1) make with the coordinate axes?

b) What is the projection matrix P onto that vector?

Answer 1

The vector (1,1,...,1) makes the same angle θ with each of the coordinate axes in n dimensions, which is found using the dot product formula. For the projection matrix P onto this vector, we use the formula involving the transpose of the vector, resulting in each element of the matrix being 1/n.

Step-by-step explanation:

Angle with Coordinate Axes

In n dimensions, the vector (1,1,...,1) makes the same angle θ with each of the coordinate axes. To find this angle θ, we can use the dot product formula between the vector and one of the unit coordinate vectors, say (1,0,...,0). The dot product is 1, and the magnitude of both vectors is √n. Applying the formula for the dot product, we get cos(θ) = 1 / √n, which gives us θ = cos-1(1 / √n).

Projection Matrix

For the projection matrix P onto the vector (1,1,...,1), we use the formula P = (v vT) / (vT v), where v is our vector and vT is its transpose. With our vector having all ones, vT v is n. Each element of v vT is 1, so P has each element equal to 1/n.