Page 107 - A Course in Linear Algebra with Applications

4.1: Examples of Vector Spaces 91

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segments IU and V representing the vectors A and B, com-
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plete the parallelogram formed by the lines U and IV; the
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the diagonal W will represent the vector A + B.
An equivalent formulation of this is the triangle rule,
which is encapsulated in the diagram which follows:

I A

Note that this diagram is obtained from the parallelogram by
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deleting the upper triangle. Since V and U W are parallel
line segments of equal length, they represent the same vector
B.
There is also a geometrical interpretation of the rule of
3
scalar multiplication in R . As before let A in R 3 be repre-
sented by the line segment joining I(tii, U2, u^) to U(tti + ai,
U2 + Q2) ^3 + 03). Let c be any scalar. Then cA is represented
by the line segment from (u\, U2, U3) to (u\ + cax, U2 + ca2,
U3 + CGS3). This line segment has length equal to \c\ times the
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length of IU, while its direction is the same as that of U if
c > 0, and opposite to that of U if c < 0.
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Of course, there are similar geometrical representations
of vectors in R 2 by line segments drawn in the plane, and in
R 1 by line segments drawn along a fixed line. So our first
examples of vector spaces are familiar objects if n < 3.
Further examples of vector spaces are obtained when the
field of real numbers is replaced by the field of complex num-
bers C: in this case we obtain
n
C ,