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

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In summary, orthogonal projection is a geometric transformation used to project points or objects onto a lower-dimensional space while maintaining their distance and angle. It is commonly used in mathematics, engineering, and computer graphics. Its purpose in scientific research is to simplify complex data and make it easier to analyze, and it differs from other projections by preserving the size and shape of objects and their angles. The D&K Example 1.5.3 in "Fundamentals of Matrix Computations" demonstrates its use in solving linear least squares problems. In real-world applications, orthogonal projection is used in computer graphics, computer vision, engineering, statistics, and data compression.
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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

Last edited:
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 $\mathbb{R}^{n}$.

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.

## What is the concept of orthogonal projection?

Orthogonal projection is a geometric transformation that projects a point or object onto a lower-dimensional space while preserving the distance and angle between points. It is commonly used in mathematics, engineering, and computer graphics.

## What is the purpose of using orthogonal projection in scientific research?

Orthogonal projection is often used in scientific research to simplify complex data or objects and make them easier to analyze. It can also be used to visualize three-dimensional data in two dimensions, making it easier to interpret and understand.

## How is orthogonal projection different from other types of projections?

Unlike other types of projections, such as perspective or oblique projection, orthogonal projection maintains the relative size and shape of objects and preserves the angles between them. This makes it useful for precise measurements and analysis.

## What is the D&K Example 1.5.3 in relation to orthogonal projection?

The D&K Example 1.5.3 refers to a specific example in the textbook "Fundamentals of Matrix Computations" by David S. Watkins. This example illustrates the use of orthogonal projection in solving a linear least squares problem, where the goal is to find the best-fit line for a set of data points.

## How is orthogonal projection used in real-world applications?

Orthogonal projection has many practical applications, such as in computer graphics for creating 3D models, in computer vision for object recognition and tracking, and in engineering for analyzing stress and strain in structures. It is also commonly used in statistics for data analysis and in data compression for reducing the size of large datasets.

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