Page 22 - Linear Algebra Done Right

Chapter 1. Vector Spaces
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Having dealt with addition in F , we now turn to multiplication. We
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could define a multiplication on F in a similar fashion, starting with
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two elements of F and getting another element of F by multiplying
corresponding coordinates. Experience shows that this definition is not
useful for our purposes. Another type of multiplication, called scalar
multiplication, will be central to our subject. Specifically, we need to
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define what it means to multiply an element of F by an element of F.
We make the obvious definition, performing the multiplication in each
coordinate:
a(x 1 ,...,x n ) = (ax 1 ,...,ax n );
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here a ∈ F and (x 1 ,...,x n ) ∈ F .
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In scalar multiplication, Scalar multiplication has a nice geometric interpretation in R .If
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we multiply together a a is a positive number and x is a vector in R , then ax is the vector
scalar and a vector, that points in the same direction as x and whose length is a times the
getting a vector. You length of x. In other words, to get ax, we shrink or stretch x by a
may be familiar with factor of a, depending upon whether a< 1or a> 1. The next picture
the dot product in R 2 illustrates this point.
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or R , in which we
multiply together two
vectors and obtain a
(3/2)x
scalar. Generalizations
x
of the dot product will
(1/2)x
become important
when we study inner
Multiplication by positive scalars
products in Chapter 6.
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You may also be If a is a negative number and x is a vector in R , then ax is the vector
familiar with the cross that points in the opposite direction as x and whose length is |a| times
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product in R , in which the length of x, as illustrated in the next picture.
we multiply together
two vectors and obtain
x
another vector. No
useful generalization of
this type of (−1/2) x
multiplication exists in
higher dimensions. (−3/2)x

Multiplication by negative scalars