The Algebra of Vector Operations

1.3 The Algebra of Vector Operations
In the previous two sections we looked at row and column vectors, matrices, functions
and directed line segments all of which are called vectors. For each of these types of
vectors we saw how addition, negation, subtraction and multiplication by a number was
defined.
It turns out that many of the familiar algebraic properties of numbers have analogues for
the vector operations. Eight of these properties are often regarded as the primary
properties. These are listed here. Here u, v and w are vectors of one of the types
discussed in the previous two sectors and s and t are any two numbers.
(1)
u+v = v+u
(commutative property)
(2)
(u + v) + w = u + (v + w)
(associative property)
(3)
u+0 = u
(0 is the zero vector)
(4)
u + (-u) = 0
(0 is the zero vector)
(5)
s(u + v) = sv + su
(distributive property)
(6)
(s + t)u = su + tu
(another distributive property)
(7)
s(tu) = (st)u
(another associative property)
(8)
1u = u
(1 is the number 1)
Most of these properties are easy to prove for row and column vectors, matrices or
functions. To prove them for directed line segments, it is easier to use the
correspondence between directed line segments and ordered pairs or triples.
u
Example 1. Prove property 6 for ordered pairs u = u12.
u1
(s + t)u1
su1 + tu1
su1
tu1
u1
u1
(s + t)u = (s + t)u  = (s + t)u  = su + tu  = su  + tu  = s u  + t u  = su + tu.
2
2
2
2
2
2
a a12 
Example 2. Prove property 6 for 22 matrices A =  a11
.
21 a22 
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2
2
a11 a12
(s+t)a11 (s+t)a12
sa11+ta11 sa12+ta12
(s + t)A = (s + t) a a  =  (s+t)a (s+t)a  =  sa +ta sa +ta 
21
22
21
22
21
21
22
22
sa11 sa12   ta11 ta12 
a11 a12 
a11 a12 



=  sa sa  +  ta ta  = s  a a  + t  a a  = sA + tA.
21
22
21
22
21
22
21
22
Example 3. Prove property 6 for functions f with domain S.
Let x be any element of S. [(s + t)f](x) = (s + t)[f(x)] = s[f(x)] + t[f(x)] = (sf)(x) + (tf)(x) =
[(sf) + (tf)](x). This implies (s + t)f = sf + tf.
Note that once one proves properties (1) - (8) for functions, it follows that they hold for
row and column vectors and matrices since row and column vectors and matrices are
functions on their index set. It also follows that they hold for directed line segments
since directed line segments can be identified with ordered pairs and triples of numbers.
The reason the above eight properties are regarded as special is because many other
familiar properties can be deduced from them. For example, here are five properties that
follow from (1) – (8).
Proposition 1. If a collection of objects V with addition, negation and multiplication by
a number satisfies (1) – (8) then the following hold for all u and v in V and numbers t.
(9)
(10)
t0 = 0
0u = 0
(11)
(12)
(13)
(-1)u = - u
u + (v – u) = v
- (u + v) = (- u) + (- v)
(0 is the zero vector)
(0 on the left is the number 0 and 0 on the right is
the zero vector)
(v – u) is what you add to u to get v
Proof. As an illustration we prove (10) and leave the rest as exercises.
0u = (0 + 0)u
by (3) for numbers
= 0u + 0u
by (6)
Add the negative of 0u to both sides:
0u + (- (0u)) = [0u + 0u] + (- (0u))
= 0u + [0u + (- (0u))]
by (2)
0 = 0u + 0
by (4)
0 = 0u
by (3)
//
Problem 1. Use (1) – (8) to prove (9). Do this by filling in the blanks in the following proof.
0 = (0 + 0)
= 0 + 0
Add the negative of 0 to both sides:
0 + (- (0)) = [0 + 0] + (- (0))
= 0 + [0 + (- (0))]
0 = 0 + 0
0 = 0
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by ______
by ______
by ______
by ______
by ______
The operation of transposing for row and column vectors and matrices also has some
useful properties.
Proposition 2. If u and v are row and column vectors or matrices with the same size then
(uT)T = u
(u + v)T = uT + vT
(u - v)T = uT - vT
(cu)T = cuT
(-u)T = - (uT)
(14)
(15)
(16)
(17)
(18)
u
v
Proof. As an illustration we prove (15) if u = u12 and v = v12 are ordered pairs and leave
u
v
u +v
the rest for exercises. One has (u + v)T = (u12 + v12)T = u21 + v21T = (u1 + v1, u2 + v2) = (u1,
u
v
u2) + (v1, v2) = u12T + v21T = uT + vT. //
One implication of these algebraic properties is that often one can work with vectors as a
whole instead of the individual components. For example, suppose one wants to find a
3
1
3
1
2
2
1
vector u such that 3u + 5 = 5u + 7. Then 2u = 5 - 7 = -2. So u = ½ -2 = -1.
Vector spaces. There are other objects besides row and column vectors, matrices,
functions and directed line segments which one can add, negate and multiply by numbers
for which (1) – (8) hold. The term vector is applied to any type of object for which this
holds. To be more precise
Definition 1. A vector space is a collection V of objects with the following properties.
1.
One can add any two objects v and w in V to get another object v + w in V.
2.
One can multiply any object v in V by a number t to get another object tv in
V.
3.
One can negate any object v in V to get another object - v in V.
4.
There is an object 0 in V called zero.
5.
Properties (1) – (8) hold for any objects u, v and w in V and numbers s and t.
A vector is an object v in a vector space V.
In some situations we restrict the numbers to be real numbers in which case the vector
space is called a real vector space. In other situations the numbers are allowed to be any
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complex number in which case the vector space is called a complex vector space. In
either case the numbers are often called scalars to emphasize their difference from
vectors.
Subspaces. In many applications we want to only work with a subset of the vectors in a
vector space. A subspace of a vector space is a subset which is closed under addition and
multiplication by a number.
Definition 2. A subspace of a vector space V is a nonempty subset W such that if u and v
are in w and t is a number then u + v and tu are in W.
Note that the zero vector is in W in view of property (10) in Proposition 1. Since the
properties (1) – (8) hold for objects in W, it follows that a subspace of a vector space is
again a vector space.
Subspaces play an important role in the theory of linear equations. Here are two
examples, that will play a role when we discuss linear functions in the next section.
Example 4. Let W be the set of functions z = f(x) which are defined and continuous for
0  x  1. Then W is a subspace of the vector space V of all functions z = f(x) which are
defined for 0  x  1. This is because the sum of two continuous functions is continuous
and if one multiplies a continuous function by a number, the result is continuous.
Example 5. Let W be the set of functions z = f(x) which are defined and whose
derivative is defined for 0  x  1. Then W is a subspace of the vector space V of all
functions z = f(x) which are defined for 0  x  1. This is because the sum of two
functions which have a derivative again has a derivative and if one multiplies a function
which has a derivative by a number, the result has a derivative.
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