Let 𝐛=(3,1,2,1)^T. Find the closest point and the distance from 𝐛 to the following subspaces: (a

step 1 The intersection of two planes is a line. To find the distance from the origin to that line, we

Let 𝐛=(3,1,2,1)^T. Find the closest point and the distance...

Let $\mathbf{b}=(3,1,2,1)^T$. Find the closest point and the distance from $\mathbf{b}$ to the following subspaces: (a) the line in the direction $(1,1,1,1)^T ;(b)$ the plane spanned by $(1,1,0,0)^T$ and $(0,0,1,1)^T ;(c)$ the hyperplane spanned by $(1,0,0,0)^T,(0,1,0,0)^T,(0,0,1,0)^T$; (d) the hyperplane defined by the equation $x+y+z+w=0$.

Key Concepts

Euclidean Projection

The concept of Euclidean projection involves finding the point in a given subspace that is closest to a particular external point. This is done by minimizing the Euclidean distance (or norm of the difference) between the external point and any point in the subspace. The resulting point is the orthogonal projection, and it satisfies the property that the error (or difference vector) is orthogonal to the subspace.

Subspaces and Their Bases

A subspace in a vector space is a collection of vectors that is closed under vector addition and scalar multiplication. It can be described by a set of basis vectors, which are linearly independent and span the entire subspace. Determining the projection involves expressing an arbitrary vector in the subspace as a linear combination of these basis vectors.

Orthogonality and Least Squares

When projecting a vector onto a subspace, one key idea is that the residual (or error vector) between the original vector and its projection is orthogonal to the subspace. This is a central principle in methods such as the least squares approach, where the goal is to minimize the sum of the squared distances, leading directly to the concept of orthogonal projection.

Projection Formulas and Matrices

Projection formulas provide a systematic way to compute the closest point in a subspace, often using the inner product. In the case of one-dimensional subspaces (lines), the projection can be calculated directly using dot products. For higher-dimensional subspaces, orthonormal basis vectors (or other techniques like the Gram-Schmidt process) are used to construct a projection matrix, which when multiplied by the original vector yields its projection.

Distance in Euclidean Space

The distance from a point to a subspace is defined as the norm (or length) of the difference between the original point and its projection onto the subspace. This measure is calculated using the Euclidean norm and provides a quantitative evaluation of how far a point is from the subspace in the ambient space.

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