Orthogonal Projections in R2

Let W be the subspace of $\mathbb{R}^2$ spanned by (1, 1). Find the orthogonal projection $P_1$ from $\mathbb{R}^2$ to W and the orthogonal projection $P_2$ from $\mathbb{R}^2$ to the orthogonal complement of W.

In this problem, we explore the concept of orthogonal projections within the two-dimensional real number space, $\mathbb{R}^2$. At the core of this problem is the understanding of subspaces and their orthogonal complements.

The vector space $\mathbb{R}^2$ can be decomposed into any given subspace and its orthogonal complement. Here, W is a one-dimensional subspace spanned by the vector (1, 1). The task requires finding not only the projection onto W but also onto its orthogonal complement.

Orthogonal projections are essential in many fields like computer graphics, data science, and quantum mechanics where projecting data or vectors orthogonally onto a subspace simplifies analysis and computations. Finding an orthogonal projection involves using the dot product, which measures how one vector extends onto another.

Conceptually, when projecting a vector from $\mathbb{R}^2$ onto the subspace spanned by (1, 1), we are essentially dropping a perpendicular from the vector to this line. This process aids in breaking down a vector into components parallel and perpendicular to the subspace line, thus facilitating more straightforward manipulation and understanding of vector components in relation to W and its complement.

Understanding these components is critical in further studies of vector spaces, especially in applications involving least squares and optimization problems, where orthogonal projections minimize errors.