Question

Let \(\left\\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}\right\\}\) be a basis of \(\mathbb{R}^{n}\). Given \(k, 1 \leq k \leq n,\) define \(P_{k}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) by \(P_{k}\left(r_{1} \mathbf{e}_{1}+\cdots+r_{n} \mathbf{e}_{n}\right)=r_{k} \mathbf{e}_{k} .\) Show that \(P_{k}\) a linear trans- formation for each \(k\).

Short answer

\( P_{k} \) is a linear transformation as it satisfies both additivity and homogeneity.

Step-by-step solution

Define Linearity

To show that a transformation is linear, we need to check two properties: 1. **Additivity**: For any vectors \( extbf{u}, extbf{v} \in \mathbb{R}^{n} \), \( P_{k}( extbf{u} + extbf{v}) = P_{k}( extbf{u}) + P_{k}( extbf{v}) \).2. **Homogeneity**: For any scalar \( c \) and vector \( extbf{u} \in \mathbb{R}^{n} \), \( P_{k}(c extbf{u}) = c P_{k}( extbf{u}) \).

Key concepts

Additivity

Additivity is one of the two main properties that define a linear transformation. Imagine you are dealing with vectors as your ingredients, and the operation of adding them is like mixing those ingredients together. For a transformation to be additive, it means that the transformation respects this mixing process.
In mathematical terms, if we have two vectors, say \( \mathbf{u} \) and \( \mathbf{v} \), and we apply a transformation \( P_k \) to both, the result should be the same as if we first add the vectors together and then transform the new vector:

• \( P_k(\mathbf{u} + \mathbf{v}) = P_k(\mathbf{u}) + P_k(\mathbf{v}) \)

This property ensures that the transformation behaves predictably, maintaining the addition structure of the vector space. In the specific case of the projection operator \( P_k \) discussed in the exercise, additivity confirms that even if we sum two vectors, projecting this sum onto a specific basis vector still distributes across the vectors.

Homogeneity

The second keystone property of linear transformations is homogeneity. Homogeneity asserts that scaling a vector by a number and then applying the transformation is the same as applying the transformation first and then scaling the result.
Think of scaling as adjusting the knobs in a sound system. If you turn all knobs up equally, you’re maintaining the harmony. If \( c \) is a scalar and \( \mathbf{u} \) is a vector, the transformation \( P_k \) should satisfy:

• \( P_k(c \mathbf{u}) = c P_k(\mathbf{u}) \)

Why is this important? It's about preserving proportions and directions when scaling occurs. If a transformation is homogeneous, it maintains this direction and proportion within the space: all parts of the original vector are scaled consistently. In the exercise, homogeneity ensures that the projection operator \( P_k \) will apply the same scale factor \( c \) to the resulting projection of vector \( \mathbf{u} \) onto the basis vector \( \mathbf{e}_k \).

Basis of a Vector Space

A basis is like a fundamental set of building blocks for a vector space. Imagine trying to build a sculpture with unique shapes; the basis is those unique pieces you always need. Every vector in the space can be expressed as a combination of these basis vectors.
In \( \mathbb{R}^n \), a common basis is the set of standard unit vectors, but it can be any set of vectors that are:

• Linearly independent - no vector in the set can be written as a combination of others, and
• Span the space - any vector in the space can be constructed with them.

This concept is vital because it allows for simplifying problems within a vector space by reducing them to a more manageable set of vectors. In the given exercise, understanding the basis is crucial because the projection operator \( P_k \) relies on these basis vectors. It's projecting any vector onto a specific one of these basis vectors, making basis the framework for our transformation.

Projection Operator

Projection operators are tools that allow you to "filter" part of a vector, focusing on one component relative to a basis. It’s like shining a flashlight on just one aspect of a story.
A projection operator \( P_k \), given in the exercise, is a linear transformation that isolates the contribution of one basis vector of a vector \( \mathbf{x}\) in \( \mathbb{R}^n \). It retains the component aligned with a specific basis vector \( \mathbf{e}_k \) and nullifies all others:

• \( P_k( r_1 \mathbf{e}_1 + \, \cdots \, + r_n \mathbf{e}_n ) = r_k \mathbf{e}_k \)

This operation is quite significant in applications like computer graphics and solving systems of linear equations, where understanding the contribution of different components of a vector can simplify computations. In solving the exercise, confirming that \( P_k \) is a linear transformation means verifying that it fits the mold through additivity and homogeneity.