# Linear Algebra: #17 Scalar Products, Norms, etc.

*geometry*. That is, about vector spaces which have a

*distance function*. (The word “geometry” obviously has to do with the measurement of physical distances on the earth.)

So let

**V**be some ﬁnite dimensional vector space over ℜ, or ℂ. Let

**v**∈

**V**be some vector in

**V**. Then, since V ≅ ℜ

^{n}, or ℂ

^{n}, we can write

**v**= ∑ a

_{j}

**e**, (with j ranging from 1, ... , n) where {

_{j}**e**, . . . ,

_{1}**e**} is the canonical basis for ℜ

_{n}^{n}or ℂ

^{n}and a

_{j}∈ ℜ or ℂ, respectively, for all j. Then the

*length*of

**v**is defined to be the non-negative real number

||

**v**|| = √|a_{1}|^{2}+ · · · + |a_{n}|^{2}.Of course, as these things always are, we will not simply conﬁne ourselves to measurements of normal physical things on the earth. We have already seen that the idea of a complex vector space defies our normal powers of geometric visualization. Also, we will not always restrict things to ﬁnite dimensional vector spaces. For example, spaces of functions — which are almost always inﬁnite dimensional — are also very important in theoretical physics. Therefore, rather than saying that ||

**v**|| is the “length” of the vector

**v**, we use a new word, and we say that ||

**v**|| is the

*norm*of

**v**. In order to define this concept in a way which is suitable for further developments, we will start with the idea of a

*scalar product*of vectors.

**Deﬁnition**

Let F = ℜ or ℂ and let

**V**,

**W**be two vector spaces over F. A bilinear form is a mapping s :

**V × W**→ F satisfying the following conditions with respect to arbitrary elements

**v**,

**v**and

_{1}**v**∈

_{2}**V**,

**w**,

**w**and

_{1}**w**∈

_{2}**W**, and a ∈ F.

- s(
**v**+_{1}**v**,_{2}**w**) = s(**v**,_{1}**w**) + s(**v**,_{2}**w**), - s(a
**v**,**w**) = as(**v**,**w**), - s(
**v**,**w**+_{1}**w**) = s(_{2}**v**,**w**) + s(_{1}**v**,**w**) and_{2} - s(
**v**, a**w**) = as(**v**,**w**).

If

**V**=

**W**, then we say that a bilinear form s :

**V × V**→ F is symmetric, if we always have s(

**v**,

_{1}**v**) = s(

_{2}**v**,

_{2}**v**). Also the form is called positive definite if s(

_{1}**v**,

**v**) > 0 for all

**v**≠ 0.

On the other hand, if F = ℂ and f :

**V**→

**W**is such that we always have

- f(
**v**+_{1}**v**) = f(_{2}**v**_{1}) + f(_{}**v**) and_{2} - f(a
**v**) = af(**v**)

A mapping s :

**V × W**→ F such that

- The mapping given by s(·,
**w**) :**V**→ F, where**v**→ s(**v**,**w**) is semi-linear for all**w**∈**W**, whereas - The mapping given by s(
**v**, ·) :**W**→ F, where**w**→ s(**v**,**w**) is linear for all**v**∈**V**

*sesqui-linear*form.

In the case

**V = W**, we say that the sesqui-linear form is

*Hermitian*(or Euclidean, if we only have F = ℜ), if we always have s(

**v**,

_{1}**v**) = s(

_{2}**v**,

_{2}**v**). (Therefore, if F = ℜ, an Hermitian form is symmetric.)

_{1}Finally, a

*scalar product*is a positive definite Hermitian form s :

**V × V**→ F. Normally, one writes (

**v**,

_{1}**v**), rather than s(

_{2}**v**,

_{1}**v**).

_{2}Well, these are a lot of new words. To be more concrete, we have the

*inner products*, which are examples of scalar products.

**Inner products**

Thus, we are considering these vectors as column vectors, defined with respect to the canonical basis of ℂ

^{n}. Then deﬁne (using matrix multiplication)

It is easy to check that this gives a scalar product on ℂ

^{n}. This particular scalar product is called the

*inner product*.

**Remark**

One often writes

**u · v**for the inner product. Thus, considering it to be a scalar product, we just have

**u · v**= <

**u**,

**v**>.

This inner product notation is often used in classical physics; in particular in Maxwell’s equations. Maxwell’s equations also involve the “vector product”

**u × v**. However the vector product of classical physics only makes sense in 3-dimensional space. Most physicists today prefer to imagine that physical space has 10, or even more — perhaps even a frothy, undefinable number of — dimensions. Therefore it appears to be the case that the vector product might have gone out of fashion in contemporary physics. Indeed, mathematicians can imagine many other possible vector-space structures as well. Thus I shall dismiss the vector product from further discussion here.

**Deﬁnition**

A real vector space (that is, over the ﬁeld of the real numbers ℜ), together with a scalar product is called a

*Euclidean*vector space. A complex vector space with scalar product is called a

*unitary*vector space.

Now, the basic reason for making all these definitions is that we want to define the length — that is the norm — of the vectors in

**V**. Given a scalar product, then the norm of

**v**∈

**V**— with respect to this scalar product — is the non-negative real number

||

**v**|| = √<**v**,**v**> .More generally, one defines a norm-function on a vector space in the following way.

**Deﬁnition**

Let

**V**be a vector space over ℂ (and thus we automatically also include the case ℜ ⊂ ℂ as well). A function || · || :

**V**→ ℜ is called a norm on

**V**if it satisfies the following conditions.

- ||a
**v**|| = |a| ||**v**|| for all**v**∈**V**and for all a ∈ ℂ, - ||
**v**+_{1}**v**|| ≤ ||_{2}**v**|| + ||_{1}**v**|| for all_{2}**v**∈**V**(the triangle inequality), and - ||
**v**|| = 0 ⇔**v**=**0**.

**Theorem 46 (Cauchy-Schwarz inequality)**Let

**V**be a Euclidean or a unitary vector space, and let ||

**v**|| = √<

**v**,

**v**> for all

**v**∈

**V**. Then we have

|<

**u, v**>| ≤ ||**u**|| · ||**v**||for all

**u**and

**v**∈

**V**. Furthermore, the equality |<

**u, v**>| ≤ ||

**u**|| · ||

**v**|| holds if, and only if, the set {

**u, v**} is linearly

*dependent*.

*Proof*

It suffices to show that |<

**u, v**>|

^{2}≤ <

**u, u**><

**v, v**>. Now, if

**v**=

**0**, then — using the properties of the scalar product — we have both <

**u, v**> = 0 and <

**v, v**> = 0. Therefore the theorem is true in this case, and we may assume that

**v**≠

**0**. Thus <

**v, v**> > 0. Let

which gives the Cauchy-Schwarz inequality. When do we have equality?

If

**v**=

**0**then, as we have already seen, the equality |<

**u, v**>| ≤ ||

**u**|| · ||

**v**|| is trivially true. On the other hand, when

**v**≠

**0**, then equality holds when <

**u**− a

**v**,

**u**− a

**v**> = 0. But since the scalar product is positive definite, this holds when

**u**− a

**v**=

**0**. So in this case as well, {

**u, v**} is linearly dependent.

**Theorem 47**Let

**V**be a vector space with scalar product, and define the non-negative function || · || :

**V**→ ℜ by ||

**v**|| = √<

**v**,

**v**> . Then || · || is a norm function on

**V**.

*Proof*

The ﬁrst and third properties in our definition of norms are obviously satisfied. As far as the triangle inequality is concerned, begin by observing that for arbitrary complex numbers z = x + yi ∈ ℂ we have

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