between the planets Vectoria and Vectoria’

This page is a sub-page of our page on Stories.

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Related KMR-pages:

• Vectors
• Linear Transformations
• Matrix Algebra
• Determinants
• Eigenvalues and Eigenvectors
• Diagonalization of Quadrics
• Singular Values Decomposition
• The Least Squares Method
• The Pseudoinverse

• Representation and Reconstruction of linear transformations.
• Setting up a Linear Space Probe for visually exploring linear transformations.
• The Rank-Nullity theorem.
• Shift of Basis for a linear transformation.

• Non-Euclidean Geometry
• Dimension
• Einstein for Flatlanders
• Einstein for Linelanders

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Other related sources of information:

• Flatland – the movie 2008, based on the novel Flatland by Edwin A. Abbott from 1884.
• Flatland – the limit of our consciousness
• A Wrinkle In Time, 2018 Disney film based on the novel by Madeleine L’Engle from 1962.
• How mathematicians are storytellers and numbers are the characters,
by Marcus du Satoy in The Guardian, 23 January 2015. 
• Taxén, G., Naeve, A. (2001), CyberMath – Exploring Open Issues in VR-based Learning, SIGGRAPH 2001 Educators Program, In SIGGRAPH 2001 Conference Abstracts and Applications, pp. 49-51.
• Taxén, G., Naeve, A., (2001), CyberMath: A Shared Virtual Environment for Mathematics Exploration , Proceedings of the 20:th World Conference on Open Learning and Distance Education (ICDE-2001), Düsseldorf, Germany, April 1-5, 2001.

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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The Tesseract or HyperCube

Tesseract – 6 rotations:

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The Triact or HyperSquare that explains the HyperCube (= Tesseract):

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BACKGROUND AND HISTORY OF THE LINEAR WAR

Paradise: In the beginning of Vectoria, there was the free, additive group of vectors, happily adding themselves according to the rules of:

• commutativity : \;\;\;\; u + v \, = \, v + u

• associativity: \;\;\;\; u + (v + w) \, = \, (u + v) + w

• existence of additive unit: \;\;\;\; 0 + v \, = \, v

• existence of additive inverse: \;\;\;\; v + -v = 0

The snake: Enter the snake that tempts the happy ‘vector groupies’ in \, V \, to eat from the tree of knowledge regarding the action of a group (of scalars) on a set (of vectors). In this way, the vectors learn how to shrink and expand themselves through multiplication by scalars from a nearby ring or field \, \mathbb {R} . The snake performs this temptation by introducing the scalar action mapping:

{\mathbb{R}\times V \ni (\lambda, v) \, \mapsto \, \lambda v \in V}.

The exclusion from paradise: This creates the knowledge among the vectors of the power of linear combinations. “And they saw that they were linearly combinable”. This insight leads to the rise of gangs of power-hungry vectors that try to control the free abelian group of additive vectors and turn it into a vector space by assigning to each vector the coefficients that express it in various linear combinations. The members of each gang fight among themselves to become as few as possible and still get the job done (just as corporations go through great trouble to eliminate redundant personnel). The surviving gang members are called basis vectors, and such a “minimal gang” is called a basis.

The sequence of coefficients of the unique linear combination that such a basis \, B \, can produce for each vector \, v \in V \, is called the coordinates of the vector \, v \, in the basis \, B .

Notation: The coordinates of the vector \, v \, in the basis \, B \, are denoted by [ \, v \, ]_B.

Human consequences: The citizens of Vectoria are coerced into being dominated by basis-gangs and get their sign of subjugation to a certain basis-gang tattooed into their skin in terms of their (unique) coordinates for this basis. Fights between the basis-gangs over the recruitment of citizens lead to subjugated vectors being converted from being coordinatized by one basis-gang into being coordinatized by another basis-gang, and being re-tattooed with another coordinate-tuple through change of basis.

Maximizing the efficiency of coordinatization: Some basis-gangs find it easier to coordinatize their subordinates than others. In fact, the computation of the coordinates become most simple if the corresponding basis-gang consists of orthonormal vectors. Such vectors are orthogonal, which means that their inner product with each other is zero, and that they have unit length.

For such an orthonormal basis, each coordinate of a vector can be computed directly by taking the inner product of this vector with the corresponding basis vector – instead of having to solve an entire system of linear equations – in order to obtain the value of each separate coordinate.

Using the inner product of \, V , non-orthogonal basis-gangs therefore try to reconfigure themselves to become orthonormal by the use of orthogonal projection. In particular, they make use of the Gram-Schmidt orthonormalization process to split a set of vectors into perpendicular (= orthogonal) components, and re-scale the length of each of these components to unity.

Space travel: The basis-gangs fight for domination among themselves until only one gang survives. This basis-gang pronounces itself as the basis-kings \, B . Having achieved domination on the initial planet \, V , the basis-kings decide to colonize another planet \, V' . They travel to \, V' \, and find that it is ruled by its own set of basis-kings called \, B' . In order to survive in \, V' , the basis-kings from \, V \, must submit to being coordinatized by \, B' , and they each get their coordinates tattooed into their skin.

Soon the immigrant basis-kings from \, V \, are plotting to overthrow the ruling basis-kings of \, V' . In order to build an army to back their case, they have to ship their subjugates from \, V \, over to \, V' . For this purpose they construct a special spaceship \, M \, – called a matrix – by aligning their own \, B' -coordinates in columns and send the matrix ship back to \, V . The basis-kings of \, V \, then command the citizens of \, V \, to line up at the space port – to be transported to \, V' \, through matrix multiplication by \, M . This means that each citizen of \, V \, gets its coordinate vector multiplied (from the left) by the matrix \, M \, and the resulting \, B' -coordinate vector determines its position as an ‘immigrant’ in \, V' .

Once in \, V' , the citizens of \, V \, find themselves in the linear subspace of \, V' \, spanned by the new positions of their own basis-kings from \, V \, in the hostile world \, V' .

Diagonalization = ?

War: It turns out that the base-kings from \, V \, did not choose their positions wisely in \, V' , because the subspace which they span does not cover the whole of \, V' . Hence they fail to take over \, V' \, directly, but they manage to take over the subspace of \, V' \, which they span.

Conflict among the immigrant kings: An internal conflict emerges among the immigrant basis-kings from \, V . There are now too many kings for unique coordinatization of their new subjects, since each king no longer commands his own separate dimension, which is independent from all the others – as they used to do back in \, V .

Surviving the space trip – or getting killed by the kernel: The surplus of immigrant kings becomes apparent when it is discovered that some of their subjects from \, V \, were killed
(i.e., mapped to zero) during their trip from \, V \, to \, V' . The investigation into these deaths is performed by computing the kernel of the space-travel-mapping described by the matrix \, M , which, from now, we will refer to as the mapping \, M: V \, \rightarrow \, V' .

The kernel of this map \, {\mathbb{R}}^3 \, \rightarrow {\mathbb{R}}^3 \, is the red line determined interactively in CyberMath:

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NOTE: The CyberMath program (CyberMath – Exploring Open Issues in VR-based Learning , CyberMath: A Shared Virtual Environment for Mathematics Exploration) was constructed in 1999 and is no longer maintained.

Instead there is an application called Exploring the action of a linear map.gcf written in the Graphing Calculator in which a linear map can be interactively explored. It can be
navigated by using the Free Viewer available there. Here you will find some films that show the action of this modern version of CyberMath:
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As a result of this conflict, some members of the basis-kings of \, V \, are eliminated (using the Gram-Schmidt process of orthonormalization), until the remaining kings become an orthonormal basis for the linear subspace that they span.

The number of surviving basis-kings from \, V \, is equal to the rank of the matrix M, which is equal to the dimension of the subspace in \, V' \, spanned by the initial basis-king immigrants from \, V.

Since the remaining immigrant basis-kings form a basis for their immigrant subjects, they are now able to coordinatize their immigrants in a unique way, using fewer coordinates than the dimensions of \, V' . In fact they need as many coordinates as the rank of the immigrant linear subspace. And since the remaining immigrant basis-kings form an orthonormal basis for this subspace, these “immigrant coordinates” can be computed by taking their inner products with the corresponding basis-vectors for the immigrant subspace.

Traveling back to Vectoria to get help: The remaining immigrant basis-kings now form an orthogonal basis of their subspace in \, V' . They now make use of this to organize a trip back to their home planet \, V \, by constructing the pseudoinverse \, P: V' \rightarrow V \, of the space-travel mapping \, M: V \rightarrow V' .

Traveling from \, V' \, to \, V \, with this pseudo-inverse, these vectors end up in the subspace of \, V \, that is orthogonal to the kernel of the mapping \, M: V \rightarrow V' . For each immigrant vector \, v' \, in \, V' , this subspace contains the shortest vector \, v \, in \, V \, that is mapped by \, M \, to \, v' , i.e., \, M(v) = v' .

Least squares fit

Finding a wormhole on \, V \, through the kernel of \, M \, : The study of the kernel of \, M \, called \, \ker M , leads to an exiting discovery. An expedition of brave explorers is sent into the black hole of \, \ker M , and after quite a turbulent experience they find themselves on a new planet Vectoria”. Here they discover the missing citizens of \, V \, that were thought to have been killed by \, M \, in the trip from \, V \, to \, V' . In contrast, it turns out that they were in fact transported to the planet \, V'' \, through a wormhole \, W \, that connects the planets \, V \, and \, V''. The entrance to \, W \, from \, V \, is \, \ker M .

In order to get everybody safely back from V'' to V the explorers now set out to discover an entrance to W from V''. They first experiment with direct mappings V'' \rightarrow V but they are unsuccessful, losing a couple of brave “kernel divers” in the process.

Then one of the researchers has a bright idea: “Why not try with mappings V'' \rightarrow V' and see if we can find a kernel that transports us back from V'' to V, just as the kernel of M: V \rightarrow V' transported us from V to V'' ?”

The explorers set out to try this new idea, and construct a mapping \, M'': V'' \rightarrow V' \, with the same image space as the mapping \, M: V \rightarrow V' . This can be achieved because, fortunately, the explorers have brought with them the coordinates in \, V' \, of their immigrant basis-kings when they went from \, V \, to \, V'' \, through the wormhole \, W . After experimenting with some suitable kernels for \, M'' \, and losing a few more divers of \, \ker M'' , one of them suddenly returns! It turns out that the corresponding \, \ker M'' leads back to \, V , and by using the \, \ker M \, entrance to \, W , the diver could go back to \, V'' \, and tell the others that a passage back to \, V \, has been discovered.

Gaming situation: On their search for the way back to earth, the crew of the spaceship Enterprise land on the planet Vectoria’ in the subspace occupied by the immigrated vectorians who greet them as friends and treat them with hospitality. The crew members then learn about the ongoing war between Vectoria and Vectoria’. They also learn that the Vectorians can help them to get closer to earth, but only if they can win the war against Vectoria’. In order to do so, the vectorians must have help to expand the subspace of Vectoria’ that they are presently occupying in order to span the whole of Vectoria’ and challenge its ruling basis-kings. The crew members can help to do that by placing vectorians, who are line-men, in dimensions of which they are not yet aware. The problem is that no one knows quite how many dimensions that Vectoria’ contains. Clues to this fact are obtained by observing various orthogonal projections of what goes (= transformations) in Vectoria’ onto the subspace occupied by the immigrants from Vectoria.

The strategy to win the war: The immigrants from Vectoria are trapped in their own subspace of Vectoria’. Like all the subjects of Vectoria’ they are subject to transformations dictated by the basis-kings of Vectoria’. In order to gain control of \, V' , the immigrants from \, V \, must figure out a way to coordinatize these super-transformations.

They first study the restriction of a mapping \, V \rightarrow V' \, to the subspace spanned by the remaining immigrant kings of \, V . The number of these is given by the rank of \, M . The simplest description of a transformation will take place in a basis whose vectors only change their length when they are transformed, but whose directions remain unchanged by the transformation. Such vectors are called eigenvectors of the transformation, and the scale factor that changes the length of each eigenvector is called the corresponding eigenvalue of the transformation [Ref].

The immigrant kings try to find eigenvectors of the \, V' \, \rightarrow \, V' \, transformations proclaimed by the rulers of \, V' . Such an eigenvector spans a one-dimensional invariant subspace of the transformation. Failing to fill up (= span) their own subspace of \, V' \, with such eigenvectors, the immigrant kings then try to find invariant subspaces of dimension higher than one. If the \, V' \, \rightarrow \, V' \, transformation is not semi-simple, they end up with Jordan boxes where each box is spanned by a chain of generalized eigenvectors.

Having managed to compute the invariant subspace structure of their own subspace of \, V', the immigrant kings now set out to expand their understanding of the super-structure that surrounds them by adding line-men in different (unknown) dimensions. If no one-dimensional invariant subspaces are found, the immigrant kings start working with plane-men (= 2space-men), who can discern invariant subspaces of dimension two. Failing to find any of these, the immigrant kings proceed to send out 3space-men, who can discern invariant subspaces of dimension three, and so on.

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