Orthogonal Projection .... .... D&K Example 1.5.3 .... ....

[Math Amateur]

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of Example 1.5.3 ...

Duistermaat and Kolk"s Example 1.5.3 reads as follows:View attachment 7707In the above example we read the following:

" ... ... Then the orthogonal projection $$f: \mathbb{R}^n \rightarrow \mathbb{R}^p$$ with $$f(x) = ( x_1, \ ... \ ... x_p )$$ ... ... "My question regards D&K's understanding of an orthogonal projection ... ...Wikipedia describes a projection (orthogonal?) as follows:

" In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such
that P ² = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged."How do we square D&K's orthogonal projection f with Wikipedia's definition of a projection ... ?

Indeed can someone please explain the nature of an orthogonal projection and how D&K's function f qualifies as such ... ...Help will be much appreciated ...

Peter

 

[GJA]

Hi, Peter.

You want to think of $\mathbb{R}^{p}$ as being a subspace of $\mathbb{R}^{n}$ and to think of $f(x) = (x_{1},\ldots , x_{p}, \underbrace{0,\ldots,0}_{n-p}).$

To see that $f(x)$ is an orthogonal projection, note that its range (=$\mathbb{R}^{p}$) and its null space (=$\mathbb{R}^{n-p}$) are orthogonal with respect to the standard inner product on $\mathbb{R}^{n}$.