Let the square matrix P be idempotent, meaning that P^2=P....

Let the square matrix $P$ be idempotent, meaning that $P^2=P$. (a) Prove that $\mathbf{w} \in \operatorname{img} P$ if and only if $P \mathbf{w}=\mathbf{w}$. (b) Show that img $P$ and $\operatorname{ker} P$ are complementary subspaces, as defined in Exercise 2.2.24, so every $\mathbf{v} \in \mathbb{R}^n$ can be uniquely written as $\mathbf{v}=\mathbf{w}+\mathbf{z}$ where $\mathbf{w} \in \operatorname{img} P, \mathbf{z} \in \operatorname{ker} P$.

Key Concepts

Direct Sum and Complementary Subspaces

A direct sum of subspaces refers to a situation where a vector space can be uniquely expressed as a sum of vectors taken from each subspace, with no overlap other than the zero vector. Two subspaces are complementary if every vector in the overall space can be written as a unique sum of one vector from each subspace. In the case of an idempotent matrix, its image and kernel naturally form complementary subspaces, ensuring a unique decomposition of any vector as the sum of a part lying in the image and a part lying in the kernel.

Image (or Range) of a Linear Transformation

The image of a linear transformation, also known as its range, is the set of all output vectors that can be obtained from applying the transformation to vectors in the domain. For an idempotent matrix considered as a projection operator, the image is the subspace of vectors that remain unchanged under the action of the matrix, thereby characterizing the part of the space onto which the projection is made.

Kernel (or Null Space) of a Linear Transformation

The kernel or null space of a linear transformation is the set of all vectors that are mapped to the zero vector by the transformation. In the context of projection operators, the kernel represents the subspace of vectors that are completely 'lost' or annihilated by the projection. This concept is essential in understanding how a transformation breaks up the space into distinct, non-overlapping components.

Projection Operator

A projection operator is a linear transformation that maps vectors onto a subspace such that applying the transformation twice yields the same result as applying it once. In other words, after projecting a vector, further projections do not change the vector. This operator naturally leads to a splitting of the space into two parts: one that is invariant under the operator and one that is annihilated by it.

Idempotent Matrix

An idempotent matrix is a matrix P that satisfies the condition P^2 = P. This key property means that when the matrix is applied twice to any vector, the result is the same as applying it once. In the context of linear algebra, idempotency is significant because it is a defining property of projection operators, which essentially 'project' vectors onto a certain subspace.

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