Page 21 - Linear Algebra Done Right

Definition of Vector Space

0 x 7
A vector
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Whenever we use pictures in R or use the somewhat vague lan-
guage of points and vectors, remember that these are just aids to our
understanding, not substitutes for the actual mathematics that we will
develop. Though we cannot draw good pictures in high-dimensional
spaces, the elements of these spaces are as rigorously defined as ele-
√
5
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ments of R . For example, (2, −3, 17,π, 2) is an element of R , and we
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may casually refer to it as a point in R or a vector in R without wor-
5
rying about whether the geometry of R has any physical meaning.
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Recall that we defined the sum of two elements of F to be the ele- Mathematical models
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ment of F obtained by adding corresponding coordinates; see 1.1. In of the economy often
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the special case of R , addition has a simple geometric interpretation. have thousands of
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Suppose we have two vectors x and y in R that we want to add, as in variables, say
the left side of the picture below. Move the vector y parallel to itself so x 1 ,...,x 5000 , which
that its initial point coincides with the end point of the vector x. The means that we must
sum x + y then equals the vector whose initial point equals the ini- operate in R 5000 . Such
tial point of x and whose end point equals the end point of the moved a space cannot be dealt
vector y, as in the right side of the picture below. with geometrically, but
the algebraic approach
works well. That’s why
our subject is called
x x
y linear algebra.

0 0
x + y
y

The sum of two vectors
Our treatment of the vector y in the picture above illustrates a standard
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philosophy when we think of vectors in R as arrows: we can move an
arrow parallel to itself (not changing its length or direction) and still
think of it as the same vector.