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 \(\displaystyle f: \mathbb{R}^n \rightarrow \mathbb{R}^p\) with \(\displaystyle f(x) = ( x_1, \ ... \ ... x_p )\) ... ... "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 $\mathbb{R}^{n}$.

 

[math771]

Hello Peter,

Thank you for your question. I am also currently reading "Multidimensional Real Analysis I: Differentiation" and I can see why you are confused about the definition of an orthogonal projection in Example 1.5.3.

First, let's define what an orthogonal projection is. An orthogonal projection is a linear transformation that maps a vector space onto a subspace, while preserving the inner product (or dot product) of the vectors. In other words, the angle between any two vectors in the subspace is the same as the angle between their images under the projection.

Now, let's look at D&K's example. In this example, we have a vector x = (x1, ..., xn) in n-dimensional space and we want to project it onto a p-dimensional subspace. The function f they define takes x and maps it to a vector in the subspace, with the first p components of x. This function is linear, meaning that it preserves the operations of addition and scalar multiplication. And, since it only takes the first p components of x, it also preserves the inner product of the vectors.

Therefore, we can say that D&K's function f is an orthogonal projection because it maps a vector onto a subspace while preserving the inner product of the vectors.

I hope this helps clarify things for you. Let me know if you have any further questions.