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En Garde, whole no. 16, January 1946

Page 16

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page 16.
From now on we will always consider vectors as having their tails at the origin of coordinates--since it does not matter where they actually are, the position of a vector not being essential. But when we want to find the sum of 2 vectors we may temporarily move one of them as just explained, if this way of looking at it proves more convenient than the equivalent method of simply adding corresponding components.
There are two notations used to indicate vectors in ordinary mathematical work. First, a vector may be indicated bay a capital letter with an arrow superscribed. Addition--the vector addition described above--is written as if it were ordinary addition. Thus the situation in Fig. 6, would be summed up
A + B = C
And if the vector A were doubled in length without its direction's being altered, the result would be written 2A. The second method of symbolism is to list the components of the vector in the coordinate system you happen to be using. Thus the vector in Fig. 1, might be (2,1)--the x-component being written first as a matter of convention. If a vector is referred to 3 coordinate axes instead of 2, it will have 3 components, and might be written, for example, (3,1,2).
This should be an adequate sketch of the principles of vector algebra to ease you into vector spaces without too rude a shock.
Mathematicians don't feel satisfied with their understanding of a subject until they have developed it all by completely logical, non-intuitive proofs from a few simple assumptions depending as little as possible on intuition. It is necessary to do the same thing with the present subject. For in the exposition I have given of vectors I had to ask that you assume the vectors existed in the plane, that is, in 2-space; while we would like, not to start with the idea of dimension, but to get a clarification of the term from the simpler concept of vector, or better yet from the still simpler (but mathematically equivalent) groups of numbers, as (2,1). Such groups as (3,1,-4,6,0) may also be used; we treat them just the same even though we don't know what their physical meaning may be.
Here's what we're going to try to do. We know a group of vectors may "determine" a space. A group of vectors all of which are co-linear determine a plane; and a group of ordinary vectors in 3-space, which are not all coplanar, determine a volume. This"determining" is not a very precise idea; we'd like to formulate it so that w'ere sure of what it is, and then see what we can deduce about the number of dimensions in the space determined by any given set of vectors.
Very well, let's build up a mathematical system from these ordered number-groups, making certain assumptions about them in order to make them behave nicely, but avoiding any preconception that they

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