Question

Let \(\mathbf{P}^{(m)}=\sum_{i=1}^{m}\left|e_{i}\right\rangle\left\langle e_{i}\right|\) be a projection operator constructed out of the first \(m\) orthonormal vectors of the basis \(B=\left\\{\left|e_{i}\right\rangle\right\\}_{i=1}^{N}\) of \(V .\) Show that \(\mathbf{P}^{(m)}\) projects into the subspace spanned by the first \(m\) vectors in \(B\).

Short answer

\(\mathbf{P}^{(m)}\) is a projection operator which projects an arbitrary vector in the complete space into the subspace spanned by the first \(m\) vectors in the basis \(B\).

Step-by-step solution

Definition of Projection Operator

Recognizing that a projection operator \( \mathbf{P} \) is one that when applied twice is equivalent to applying it once, i.e., \( \mathbf{P}^2 = \mathbf{P} \) .

Apply the Operator to an Arbitrary Vector

Consider an arbitrary vector \( |a\rangle \) from the complete basis \(B\), which can be written as: \( |a\rangle=\sum_{i=1}^{N} a_i |e_i\rangle \). Apply the operator \( \mathbf{P}^{(m)} \) to \( |a\rangle \), the result is: \( \mathbf{P}^{(m)}|a\rangle = \sum_{i=1}^{m} a_i |e_i\rangle \) . This demonstrates that the projection operator \( \mathbf{P}^{(m)} \) projects any arbitrary vector \( |a\rangle \) in the original complete basis \(B\) into the subspace spanned by the first \( m \) vectors in \( B \). This subspace is smaller then the space spanned by the whole basis \(B\) and thus that operation reduces the dimensions of the original vector \( |a\rangle \) from \( N \) to \( m \).

Show That the Projection Operation Is Idempotent

To confirm that \( \mathbf{P}^{(m)} \) is indeed a projection operator, show that applying it twice to an arbitrary vector is the same as applying it once. When \( \mathbf{P}^{(m)} \) is applied to \( \mathbf{P}^{(m)}|a\rangle \), the result is again \( \sum_{i=1}^{m} a_i |e_i\rangle \). Hence the idempotent property is demonstrated, and \( \mathbf{P}^{(m)} \) is a projection operator.

Key concepts

Linear Algebra

Linear algebra is a branch of mathematics that is vital in a multitude of fields such as engineering, physics, computer science, and economics. It deals with vectors, vector spaces, linear mappings, and systems of linear equations. Vectors are objects that have both magnitude and direction.

Vector spaces (or linear spaces) are collections of vectors that can be added together and multiplied by scalars (numbers) to produce another vector within the same space. A key concept in linear algebra is the transformation of vectors via linear mappings or matrices, which can affect a vector's direction, length, and position within the space.

Another fundamental concept is solving systems of linear equations, which is finding the values of variables that satisfy multiple linear equations simultaneously. This often involves techniques such as Gaussian elimination, matrix factorization, and the use of inverse matrices.

Orthonormal Basis

An orthonormal basis is a set of vectors that are both orthogonal (perpendicular) to each other and have a unit length (norm equal to one). In essence, these sets of vectors in a vector space V have the property that the inner product (or dot product in Euclidean space) of any two different vectors is zero, and the inner product of a vector with itself is one.

This property simplifies many mathematical operations such as projection, as orthogonal vectors do not interfere with each other and unit length makes the scale of projection consistent. Any vector in the space can be uniquely written as a linear combination of the basis vectors, which greatly aids in simplifying mathematical exercises and theoretical understanding.

Within linear algebra, the use of an orthonormal basis is preferred because calculations become much cleaner and more intuitive, especially when dealing with complex spaces in higher dimensions.

Vector Projection

Vector projection is a linear algebra operation wherein a vector is projected onto another vector or set of vectors, which defines a subspace. Essentially, it is like casting a shadow of one vector onto another, where the 'shadow' retains the directional properties of the original vector in the context of the subspace.

Mathematically, the projection of a vector \textbf{a} onto a vector \textbf{b} can be expressed via the formula: \[ P_{\textbf{b}}(\textbf{a}) = \frac{\textbf{a} \times \textbf{b}}{\textbf{b} \times \textbf{b}} \times \textbf{b} \. \] When we project onto a subspace rather than just a single vector, we are, in essence, reducing the dimensions while retaining as much information as possible from the original vector relative to that subspace.

In the context of the exercise at hand, the projection operator \( \mathbf{P}^{(m)} \) is projecting vectors into a subspace defined by the first \( m \) orthonormal basis vectors. The calculation simplifies due to the orthonormality, allowing for an easy understanding of the component parts of the original vector that lie within the subspace.