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En Garde, whole no. 16, January 1946
Page 17
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page 17. do resemble sets of vector-components until we have shown that they do. We'll call 'em vectors, though, right from the start. All the vectors in any one system must be required to have the same number of numbers (if you see what I mean) in order that they may be made to behave like sets of components all related to the same system of coordinates. We'll phrase the definitions and so forth so that this number of coordinates may be anything we choose; but in giving examples we'll use 4, so that typical vectors would be (1,0,0,0) and (4,-3,1,1/2). We haven't yet defined addition of vectors, so we'll do it the obvious way. The sum of vectors A & B is defined as the vector obtained by adding corresponding components of A & B. Thus if A were (0,1,5,0) and B were (1/2 -2, 0, 3), we would have A + B = (0+1/2, 1-2, 5+0, 0+3) = (1/2, -1, 5, 3) We will also define multiplication of a vector by a number in the way our previous discussion suggested. The product of vector A and number c is defined as the vector obtained by multiplying each of A 's components by c. Thus if A is (1, 0, -2, 3), then 2A = (2, 0, -4, 6) 1A = (1, 0, -2, 3) = A -2A = (-2, 0, 4, -6) 0A = (0, 0, 0, 0) If you use these definitions on a vector with only 2 components, instead of 4, you find the results agree with those obtained from the ordinary definitions used for physical vectors above. Thus these number groups we've been calling vectors do, with the definitions we've made, act like vectors. Yet the're more general, for we need not limit the number of components to 3. We can, in fact, find what would happen if we did have 4 mutually perpendicular coordinate axes, by investigating our ordered number groups of 4 numbers. III The next thing on the program is the definition of "linear dependence", a concept whose importance will become apparent later. We say , B, & C are lineraly dependent if we can find numbers a, b, & c, none of which are zero, such that aA + bB + cC = (0, 0, 0, 0) The same definition applies with obvious changes if you have a different number of vectors or a different number of components in each vector. Some examples are in order.
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page 17. do resemble sets of vector-components until we have shown that they do. We'll call 'em vectors, though, right from the start. All the vectors in any one system must be required to have the same number of numbers (if you see what I mean) in order that they may be made to behave like sets of components all related to the same system of coordinates. We'll phrase the definitions and so forth so that this number of coordinates may be anything we choose; but in giving examples we'll use 4, so that typical vectors would be (1,0,0,0) and (4,-3,1,1/2). We haven't yet defined addition of vectors, so we'll do it the obvious way. The sum of vectors A & B is defined as the vector obtained by adding corresponding components of A & B. Thus if A were (0,1,5,0) and B were (1/2 -2, 0, 3), we would have A + B = (0+1/2, 1-2, 5+0, 0+3) = (1/2, -1, 5, 3) We will also define multiplication of a vector by a number in the way our previous discussion suggested. The product of vector A and number c is defined as the vector obtained by multiplying each of A 's components by c. Thus if A is (1, 0, -2, 3), then 2A = (2, 0, -4, 6) 1A = (1, 0, -2, 3) = A -2A = (-2, 0, 4, -6) 0A = (0, 0, 0, 0) If you use these definitions on a vector with only 2 components, instead of 4, you find the results agree with those obtained from the ordinary definitions used for physical vectors above. Thus these number groups we've been calling vectors do, with the definitions we've made, act like vectors. Yet the're more general, for we need not limit the number of components to 3. We can, in fact, find what would happen if we did have 4 mutually perpendicular coordinate axes, by investigating our ordered number groups of 4 numbers. III The next thing on the program is the definition of "linear dependence", a concept whose importance will become apparent later. We say , B, & C are lineraly dependent if we can find numbers a, b, & c, none of which are zero, such that aA + bB + cC = (0, 0, 0, 0) The same definition applies with obvious changes if you have a different number of vectors or a different number of components in each vector. Some examples are in order.
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