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Chapter 5: Problem 12

Show that under the orthogonal projection of an ellipsoid lying in one subspace of euclidean space onto another subspace, all the semi-axes are decreased.

Short Answer

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Answer: Under an orthogonal projection of an ellipsoid onto a subspace of Euclidean space, all the semi-axes are either decreased or remain the same.

Step by step solution

01

Introduction to ellipsoids

An ellipsoid is a closed quadric surface that is a higher-dimensional analogue of an ellipse. In a three-dimensional Euclidean space, an ellipsoid is defined by the equation: \[(\frac{x-x_0}{a})^2 + (\frac{y-y_0}{b})^2 + (\frac{z-z_0}{c})^2 = 1\] where \((x_0, y_0, z_0)\) is the center of the ellipsoid, and \(a, b, c\) are the lengths of the semi-axes.
02

Introduction to orthogonal projections

An orthogonal projection is a linear transformation that projects a point or a set of points onto a subspace, such that the projection is orthogonal to the subspace. In simpler terms, it's a way to "flatten" an object along a certain direction, while maintaining the perpendicular distance between the original object and its projection onto the subspace.
03

Introduction to semi-axes

Semi-axes are the half-lengths of the axes of an ellipsoid. They determine the shape and size of the ellipsoid. In our case, we will be focusing on the semi-axes \(a, b, c\) as stated in Step 1.
04

Properties of orthogonal projections and ellipsoids

One of the key properties of orthogonal projections is that they preserve the angles between any two lines. This means that if we project an ellipsoid onto a subspace, the angles between the original ellipsoid and its projection will be preserved. Now, let's consider the semi-axes of the ellipsoid. The orthogonal projection will "flatten" the ellipsoid along a certain direction. Hence, the semi-axes when projecting the ellipsoid will either stay the same or decrease, depending on the direction of projection, but they will not increase.
05

Proving the statement

To prove that all semi-axes are decreased under orthogonal projection of an ellipsoid, we can use the properties stated in Step 4. Consider an ellipsoid and its orthogonal projection onto a subspace. Since the orthogonal projection preserves the angles between any two lines, and the semi-axes are only allowed to stay the same or decrease in length when projecting, the projection of the ellipsoid will always result in semi-axes which are equal to or shorter than the original ones. Hence, under orthogonal projection of an ellipsoid lying in one subspace of Euclidean space onto another subspace, all the semi-axes are decreased or remain the same.

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