The mathematical idea of a vector plays an important role in many areas of physics.
Looking at these five examples where linear algebra comes up in physics, we see that for the first three, involving “classical physics”, we have vectors placed at different points in space and time. On the other hand, the fifth example is a vector space where the vectors are not to be thought of as being simple arrows in the normal, classical space of everyday life. In any case, it is clear that the theory of linear algebra is very basic to any study of physics.
But rather than thinking in terms of vectors as representing physical processes, it is best to begin these lectures by looking at things in a more mathematical, abstract way. Once we have gotten a feeling for the techniques involved, then we can apply them to the simple picture of vectors as being arrows located at different points of the classical 3-dimensional space.
Let X and Y be sets. The Cartesian product X × Y , of X with Y is the set of all possible pairs (x, y ) such that x ∈ X and y ∈ Y .
A group is a non-empty set G, together with an operation, which is a mapping ‘ · ’ : G × G → G, such that the following conditions are satisfied.
1. For all a, b, c ∈ G, we have (a · b) · c = a · (b · c),
2. There exists a particular element (the “neutral” element), often called e in group theory, such that e · g = g · e = g, for all g ∈ G.
3. For each g ∈ G, there exists an inverse element g−1 ∈ G such that g · g−1 = g−1 · g = e.
If, in addition, we have a · b = b · a for all a, b ∈ G, then G is called an “Abelian” group.
A field is a non-empty set F, having two arithmetical operations, denoted by ‘+’ and ‘·’, that is, addition and multiplication.
Under addition, F is an Abelian group with a neutral element denoted by ‘0’. Furthermore, there is another element, denoted by ‘1’, with 1 ≠ 0, such that F \ {0} (that is, the set F, with the single element 0 removed) is an Abelian group, with neutral element 1, under multiplication. In addition, the distributive property holds:
a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c, for all a, b, c ∈ F .
The simplest example of a field is the set consisting of just two elements {0, 1} with the obvious multiplication. This is the field Z/2Z. Also, as we have seen in the analysis lectures, for any prime number p ∈ N, the set Z/pZ of residues modulo p is a field.
The following theorem, which should be familiar from the analysis lectures, gives some elementary general properties of fields.
Theorem 1. Let F be a field. Then for all a, b ∈ F, we have:
1. a · 0 = 0 · a = 0,
2. a · (−b) = −(a · b) = (−a) · b,
3. −(−a) = a,
4. (a−1)−1 = a, if a ≠ 0,
5. (−1) · a = −a,
6. (−a) · (−b) = a · b,
7. a · b = 0 ⇒ a = 0 or b = 0.
Proof. An exercise
So the theory of abstract vector spaces starts with the idea of a field as the underlying arithmetical system. But in physics, and in most of mathematics (at least the analysis part of it), we do not get carried away with such generalities.
Instead we will usually be confining our attention to one of two very particular fields, namely either the field of real numbers R, or else the field of complex numbers C.
Despite this, let us adopt the usual generality in the definition of a vector space.
A vector space V over a field F is an Abelian group — with vector addition denoted by v + w, for vectors v, w ∈ V. The neutral element is the “zero vector” 0. Furthermore, there is a scalar multiplication F × V → V satisfying (for arbitrary a, b ∈ F and v, w ∈ V):
1. a · (v + w) = a · v + a · w,
2. (a + b) · v = a · v + b · v,
3. (a · b) · v = a · (b · v), and
4. 1 · v = v for al l v ∈ V.
Scalar multiplication is multiplication in the field.
the Cartesian product, defined recursively. Given two elements
(x1 , ... , xn ) and (y1, ... , yn)
in , then the vector sum is simply the new vector
(x1 + y1, ... , xn + yn).
Scalar multiplication is
a · (x1, ... , xn) = (a · x1, ... , a · xn).
It is a trivial matter to verify that , with these operations, is a vector space over
(f + g)(x) = f (x) + g(x),
for all x ∈ [0, 1], defining the new function (f + g) ∈ C0([0, 1], for all f , g ∈ C0([0, 1], Scalar multiplication is given by
(a · f )(x) = a · f (x)
for all x ∈ [0, 1].
Let V be a vector space over a field F and let W ⊂ V be some subset. If W is itself a vector space over F , considered using the addition and scalar multiplication in V, then we say that W is a subspace of V. Analogously, a subset H of a group G, which is itself a group using the multiplication operation from G, is called a subgroup of G. Subfields are similarly defined.
Theorem 2. Let W ⊂ V be a subset of a vector space over the field F .
Then W is a subspace of V ⇔ a · v + b · w ∈ W,
for all v, w ∈ W and a, b ∈ F .
Proof. The direction ‘⇒’ is trivial.
For '⇐,’, begin by observing that 1. v + 1.w= v+w ∈ W, and a.v+0.w = a.v ∈ W, for all v, w ∈ W and a ∈ F. Thus W is closed under vector addition and scalar multiplication.
Is W a group with respect to vector addition? We have 0.v= 0 ∈ W, for v ∈ W; therefore the neutral element 0 is contained in W. For an arbitrary v ∈ W we have
v + (−1) · v = 1 · v + (−1) · v
= (1 + (−1)) · v
= 0 · v
= 0.
Therefore (−1) · v is the inverse element to v under addition, and so we can simply write (-1) .v = -v. the other axioms for a vector space can be easily checked.
The method of this proof also shows that we have similar conditions for subsets of groups or fields to be subgroups, or subfields, respectively.
1. What are subspaces in linear algebra? |
2. What is the difference between a subspace and a span in linear algebra? |
3. How do you determine if a set of vectors forms a subspace? |
4. What is the role of subspaces in solving systems of linear equations? |
5. Can a subspace have infinitely many vectors? |
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