Section VS  Vector Spaces

From A First Course in Linear Algebra
Version 2.00
© 2004.
Licensed under the GNU Free Documentation License.

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

Subsection VS: Vector Spaces

Here is one of the two most important definitions in the entire course.

Definition VS
Vector Space
Suppose that V is a set upon which we have defined two operations: (1) vector addition, which combines two elements of V and is denoted by “+”, and (2) scalar multiplication, which combines a complex number with an element of V and is denoted by juxtaposition. Then V , along with the two operations, is a vector space if the following ten properties hold.

The objects in V are called vectors, no matter what else they might really be, simply by virtue of being elements of a vector space.

Now, there are several important observations to make. Many of these will be easier to understand on a second or third reading, and especially after carefully studying the examples in Subsection VS.EVS.

An axiom is often a “self-evident” truth. Something so fundamental that we all agree it is true and accept it without proof. Typically, it would be the logical underpinning that we would begin to build theorems upon. Some might refer to the ten properties of Definition VS as axioms, implying that a vector space is a very natural object and the ten properties are the essence of a vector space. We will instead emphasize that we will begin with a definition of a vector space. After studying the remainder of this chapter, you might return here and remind yourself how all our forthcoming theorems and definitions rest on this foundation.

As we will see shortly, the objects in V can be anything, even though we will call them vectors. We have been working with vectors frequently, but we should stress here that these have so far just been column vectors — scalars arranged in a columnar list of fixed length. In a similar vein, you have used the symbol “+” for many years to represent the addition of numbers (scalars). We have extended its use to the addition of column vectors and to the addition of matrices, and now we are going to recycle it even further and let it denote vector addition in any possible vector space. So when describing a new vector space, we will have to define exactly what “+” is. Similar comments apply to scalar multiplication. Conversely, we can define our operations any way we like, so long as the ten properties are fulfilled (see Example CVS).

A vector space is composed of three objects, a set and two operations. However, we usually use the same symbol for both the set and the vector space itself. Do not let this convenience fool you into thinking the operations are secondary!

This discussion has either convinced you that we are really embarking on a new level of abstraction, or they have seemed cryptic, mysterious or nonsensical. You might want to return to this section in a few days and give it another read then. In any case, let’s look at some concrete examples now.

Subsection EVS: Examples of Vector Spaces

Our aim in this subsection is to give you a storehouse of examples to work with, to become comfortable with the ten vector space properties and to convince you that the multitude of examples justifies (at least initially) making such a broad definition as Definition VS. Some of our claims will be justified by reference to previous theorems, we will prove some facts from scratch, and we will do one non-trivial example completely. In other places, our usual thoroughness will be neglected, so grab paper and pencil and play along.

Example VSCV
The vector space m
Set: m, all column vectors of size m, Definition VSCV.
Equality: Entry-wise, Definition CVE.
Vector Addition: The “usual” addition, given in Definition CVA.
Scalar Multiplication: The “usual” scalar multiplication, given in Definition CVSM.

Does this set with these operations fulfill the ten properties? Yes. And by design all we need to do is quote Theorem VSPCV. That was easy.

Example VSM
The vector space of matrices, Mmn
Set: Mmn, the set of all matrices of size m × n and entries from , Example VSM.
Equality: Entry-wise, Definition ME.
Vector Addition: The “usual” addition, given in Definition MA.
Scalar Multiplication: The “usual” scalar multiplication, given in Definition MSM.

Does this set with these operations fulfill the ten properties? Yes. And all we need to do is quote Theorem VSPM. Another easy one (by design).

So, the set of all matrices of a fixed size forms a vector space. That entitles us to call a matrix a vector, since a matrix is an element of a vector space. For example, if A,B M3,4 then we call A and B “vectors,” and we even use our previous notation for column vectors to refer to A and B. So we could legitimately write expressions like

u + v = A + B = B + A = v + u

This could lead to some confusion, but it is not too great a danger. But it is worth comment.

The previous two examples may be less than satisfying. We made all the relevant definitions long ago. And the required verifications were all handled by quoting old theorems. However, it is important to consider these two examples first. We have been studying vectors and matrices carefully (Chapter V, Chapter M), and both objects, along with their operations, have certain properties in common, as you may have noticed in comparing Theorem VSPCV with Theorem VSPM. Indeed, it is these two theorems that motivate us to formulate the abstract definition of a vector space, Definition VS. Now, should we prove some general theorems about vector spaces (as we will shortly in Subsection VS.VSP), we can instantly apply the conclusions to both m and Mmn. Notice too how we have taken six definitions and two theorems and reduced them down to two examples. With greater generalization and abstraction our old ideas get downgraded in stature.

Let us look at some more examples, now considering some new vector spaces.

Example VSP
The vector space of polynomials, Pn
Set: Pn, the set of all polynomials of degree n or less in the variable x with coefficients from .

a0+a1x+a2x2++a nxn = b 0+b1x+b2x2++b nxn  if and only if a i = bi  for 0 i n

Vector Addition:

(a0 + a1x + a2x2 + + a nxn) + (b 0 + b1x + b2x2 + + b nxn) = (a0 + b0) + (a1 + b1)x + (a2 + b2)x2 + + (a n + bn)xn

Scalar Multiplication:

α(a0 + a1x + a2x2 + + a nxn) = (αa 0) + (αa1)x + (αa2)x2 + + (αa n)xn

This set, with these operations, will fulfill the ten properties, though we will not work all the details here. However, we will make a few comments and prove one of the properties. First, the zero vector (Property Z) is what you might expect, and you can check that it has the required property.

0 = 0 + 0x + 0x2 + + 0xn

The additive inverse (Property AI) is also no surprise, though consider how we have chosen to write it.

a0 + a1x + a2x2 + + a nxn = (a 0) + (a1)x + (a2)x2 + + (a n)xn

Now let’s prove the associativity of vector addition (Property AA). This is a bit tedious, though necessary. Throughout, the plus sign (“+”) does triple-duty. You might ask yourself what each plus sign represents as you work through this proof.

u+(v + w) = (a0 + a1x + + anxn) + (b 0 + b1x + + bnxn) + (c 0 + c1x + + cnxn) = (a0 + a1x + + anxn) + ((b 0 + c0) + (b1 + c1)x + + (bn + cn)xn) = (a0 + (b0 + c0)) + (a1 + (b1 + c1))x + + (an + (bn + cn))xn = ((a0 + b0) + c0) + ((a1 + b1) + c1)x + + ((an + bn) + cn)xn = ((a0 + b0) + (a1 + b1)x + + (an + bn)xn) + (c 0 + c1x + + cnxn) = (a0 + a1x + + anxn) + (b 0 + b1x + + bnxn) + (c 0 + c1x + + cnxn) = (u + v) + w

Notice how it is the application of the associativity of the (old) addition of complex numbers in the middle of this chain of equalities that makes the whole proof happen. The remainder is successive applications of our (new) definition of vector (polynomial) addition. Proving the remainder of the ten properties is similar in style and tedium. You might try proving the commutativity of vector addition (Property C), or one of the distributivity properties (Property DVA, Property DSA).

Example VSIS
The vector space of infinite sequences
Set: = (c 0,c1,c2,c3,) ci ,i .

(c0,c1,c2,) = (d0,d1,d2,)  if and only if ci = di  for all i 0

Vector Addition:

(c0,c1,c2,) + (d0,d1,d2,) = (c0 + d0,c1 + d1,c2 + d2,)

Scalar Multiplication:

α(c0,c1,c2,c3,) = (αc0,αc1,αc2,αc3,)

This should remind you of the vector space m, though now our lists of scalars are written horizontally with commas as delimiters and they are allowed to be infinite in length. What does the zero vector look like (Property Z)? Additive inverses (Property AI)? Can you prove the associativity of vector addition (Property AA)?

Example VSF
The vector space of functions
Set: F = ff : .
Equality: f = g if and only if f(x) = g(x) for all x .
Vector Addition: f + g is the function with outputs defined by (f + g)(x) = f(x) + g(x).
Scalar Multiplication: αf is the function with outputs defined by (αf)(x) = αf(x).

So this is the set of all functions of one variable that take a complex number to a complex number. You might have studied functions of one variable that take a real number to a real number, and that might be a more natural set to study. But since we are allowing our scalars to be complex numbers, we need to expand the domain and range of our functions also. Study carefully how the definitions of the operation are made, and think about the different uses of “+” and juxtaposition. As an example of what is required when verifying that this is a vector space, consider that the zero vector (Property Z) is the function z whose definition is z(x) = 0 for every input x.

While vector spaces of functions are very important in mathematics and physics, we will not devote them much more attention.

Here’s a unique example.

Example VSS
The singleton vector space
Set: Z = z.
Equality: Huh?
Vector Addition: z + z = z.
Scalar Multiplication: αz = z.

This should look pretty wild. First, just what is z? Column vector, matrix, polynomial, sequence, function? Mineral, plant, or animal? We aren’t saying! z just is. And we have definitions of vector addition and scalar multiplication that are sufficient for an occurence of either that may come along.

Our only concern is if this set, along with the definitions of two operations, fulfills the ten properties of Definition VS. Let’s check associativity of vector addition (Property AA). For all u,v,w Z,

u + (v + w) = z + (z + z) = z + z = (z + z) + z = (u + v) + w

What is the zero vector in this vector space (Property Z)? With only one element in the set, we do not have much choice. Is z = 0? It appears that z behaves like the zero vector should, so it gets the title. Maybe now the definition of this vector space does not seem so bizarre. It is a set whose only element is the element that behaves like the zero vector, so that lone element is the zero vector.

Perhaps some of the above definitions and verifications seem obvious or like splitting hairs, but the next example should convince you that they are necessary. We will study this one carefully. Ready? Check your preconceptions at the door.

Example CVS
The crazy vector space
Set: C = (x1,x2) x1,x2 .
Vector Addition: (x1,x2) + (y1,y2) = (x1 + y1 + 1,x2 + y2 + 1).
Scalar Multiplication: α(x1,x2) = (αx1 + α 1,αx2 + α 1).

Now, the first thing I hear you say is “You can’t do that!” And my response is, “Oh yes, I can!” I am free to define my set and my operations any way I please. They may not look natural, or even useful, but we will now verify that they provide us with another example of a vector space. And that is enough. If you are adventurous, you might try first checking some of the properties yourself. What is the zero vector? Additive inverses? Can you prove associativity? Ready, here we go.

Property AC, Property SC: The result of each operation is a pair of complex numbers, so these two closure properties are fulfilled.

Property C:

u + v = (x1,x2) + (y1,y2) = (x1 + y1 + 1,x2 + y2 + 1) = (y1 + x1 + 1,y2 + x2 + 1) = (y1,y2) + (x1,x2) = v + u

Property AA:

u + (v + w) = (x1,x2) + (y1,y2) + (z1,z2) = (x1,x2) + (y1 + z1 + 1,y2 + z2 + 1) = (x1 + (y1 + z1 + 1) + 1,x2 + (y2 + z2 + 1) + 1) = (x1 + y1 + z1 + 2,x2 + y2 + z2 + 2) = ((x1 + y1 + 1) + z1 + 1,(x2 + y2 + 1) + z2 + 1) = (x1 + y1 + 1,x2 + y2 + 1) + (z1,z2) = (x1,x2) + (y1,y2) + (z1,z2) = u + v + w

Property Z: The zero vector is …0 = (1, 1). Now I hear you say, “No, no, that can’t be, it must be (0,0)!” Indulge me for a moment and let us check my proposal.

u + 0 = (x1,x2) + (1, 1) = (x1 + (1) + 1,x2 + (1) + 1) = (x1,x2) = u

Feeling better? Or worse?

Property AI: For each vector, u, we must locate an additive inverse, u. Here it is, (x1,x2) = (x1 2, x2 2). As odd as it may look, I hope you are withholding judgment. Check:

u+(u) = (x1,x2)+(x12,x22) = (x1+(x12)+1,x2+(x22)+1) = (1,1) = 0

Property SMA:

α(βu) = α(β(x1,x2)) = α(βx1 + β 1,βx2 + β 1) = (α(βx1 + β 1) + α 1,α(βx2 + β 1) + α 1) = ((αβx1 + αβ α) + α 1,(αβx2 + αβ α) + α 1) = (αβx1 + αβ 1,αβx2 + αβ 1) = (αβ)(x1,x2) = (αβ)u

Property DVA: If you have hung on so far, here’s where it gets even wilder. In the next two properties we mix and mash the two operations.

α(u + v) = α (x1,x2) + (y1,y2) = α(x1 + y1 + 1,x2 + y2 + 1) = (α(x1 + y1 + 1) + α 1,α(x2 + y2 + 1) + α 1) = (αx1 + αy1 + α + α 1,αx2 + αy2 + α + α 1) = (αx1 + α 1 + αy1 + α 1 + 1,αx2 + α 1 + αy2 + α 1 + 1) = ((αx1 + α 1) + (αy1 + α 1) + 1,(αx2 + α 1) + (αy2 + α 1) + 1) = (αx1 + α 1,αx2 + α 1) + (αy1 + α 1,αy2 + α 1) = α(x1,x2) + α(y1,y2) = αu + αv

Property DSA:

(α + β)u = (α + β)(x1,x2) = ((α + β)x1 + (α + β) 1,(α + β)x2 + (α + β) 1) = (αx1 + βx1 + α + β 1,αx2 + βx2 + α + β 1) = (αx1 + α 1 + βx1 + β 1 + 1,αx2 + α 1 + βx2 + β 1 + 1) = ((αx1 + α 1) + (βx1 + β 1) + 1,(αx2 + α 1) + (βx2 + β 1) + 1) = (αx1 + α 1,αx2 + α 1) + (βx1 + β 1,βx2 + β 1) = α(x1,x2) + β(x1,x2) = αu + βu

Property O: After all that, this one is easy, but no less pleasing.

1u = 1(x1,x2) = (x1 + 1 1,x2 + 1 1) = (x1,x2) = u

That’s it, C is a vector space, as crazy as that may seem.

Notice that in the case of the zero vector and additive inverses, we only had to propose possibilities and then verify that they were the correct choices. You might try to discover how you would arrive at these choices, though you should understand why the process of discovering them is not a necessary component of the proof itself.

Subsection VSP: Vector Space Properties

Subsection VS.EVS has provided us with an abundance of examples of vector spaces, most of them containing useful and interesting mathematical objects along with natural operations. In this subsection we will prove some general properties of vector spaces. Some of these results will again seem obvious, but it is important to understand why it is necessary to state and prove them. A typical hypothesis will be “Let V be a vector space.” From this we may assume the ten properties of Definition VS, and nothing more. Its like starting over, as we learn about what can happen in this new algebra we are learning. But the power of this careful approach is that we can apply these theorems to any vector space we encounter — those in the previous examples, or new ones we have not yet contemplated. Or perhaps new ones that nobody has ever contemplated. We will illustrate some of these results with examples from the crazy vector space (Example CVS), but mostly we are stating theorems and doing proofs. These proofs do not get too involved, but are not trivial either, so these are good theorems to try proving yourself before you study the proof given here. (See Technique P.)

First we show that there is just one zero vector. Notice that the properties only require there to be at least one, and say nothing about there possibly being more. That is because we can use the ten properties of a vector space (Definition VS) to learn that there can never be more than one. To require that this extra condition be stated as an eleventh property would make the definition of a vector space more complicated than it needs to be.

Theorem ZVU
Zero Vector is Unique
Suppose that V is a vector space. The zero vector, 0, is unique.

Proof   To prove uniqueness, a standard technique is to suppose the existence of two objects (Technique U). So let 01 and 02 be two zero vectors in V . Then

01 = 01 + 02  Property Z for 02 = 02 + 01  Property C = 02  Property Z for 01

This proves the uniqueness since the two zero vectors are really the same.

Theorem AIU
Additive Inverses are Unique
Suppose that V is a vector space. For each u V , the additive inverse, u, is unique.

Proof   To prove uniqueness, a standard technique is to suppose the existence of two objects (Technique U). So let u1 and u2 be two additive inverses for u. Then

u1 = u1 + 0  Property Z = u1 + (u + u2)  Property AI = (u1 + u) + u2  Property AA = 0 + u2  Property AI = u2  Property Z

So the two additive inverses are really the same.

As obvious as the next three theorems appear, nowhere have we guaranteed that the zero scalar, scalar multiplication and the zero vector all interact this way. Until we have proved it, anyway.

Theorem ZSSM
Zero Scalar in Scalar Multiplication
Suppose that V is a vector space and u V . Then 0u = 0.

Proof   Notice that 0 is a scalar, u is a vector, so Property SC says 0u is again a vector. As such, 0u has an additive inverse, (0u) by Property AI.

0u = 0 + 0u  Property Z = (0u) + 0u + 0u  Property AI = (0u) + 0u + 0u  Property AA = (0u) + (0 + 0)u  Property DSA = (0u) + 0u  Property ZCN = 0  Property AI

Here’s another theorem that looks like it should be obvious, but is still in need of a proof.

Theorem ZVSM
Zero Vector in Scalar Multiplication
Suppose that V is a vector space and α . Then α0 = 0.

Proof   Notice that α is a scalar, 0 is a vector, so Property SC means α0 is again a vector. As such, α0 has an additive inverse, (α0) by Property AI.

α0 = 0 + α0  Property Z = (α0) + α0 + α0  Property AI = (α0) + α0 + α0  Property AA = (α0) + α 0 + 0  Property DVA = (α0) + α0  Property Z = 0  Property AI

Here’s another one that sure looks obvious. But understand that we have chosen to use certain notation because it makes the theorem’s conclusion look so nice. The theorem is not true because the notation looks so good, it still needs a proof. If we had really wanted to make this point, we might have defined the additive inverse of u as u. Then we would have written the defining property, Property AI, as u + u = 0. This theorem would become u = (1)u. Not really quite as pretty, is it?

Theorem AISM
Additive Inverses from Scalar Multiplication
Suppose that V is a vector space and u V . Then u = (1)u.


u = u + 0  Property Z = u + 0u  Theorem ZSSM = u + 1 + (1) u = u + 1u + (1)u  Property DSA = u + u + (1)u  Property O = u + u + (1)u  Property AA = 0 + (1)u  Property AI = (1)u  Property Z

Because of this theorem, we can now write linear combinations like 6u1 + (4)u2
as 6u1 4u2, even though we have not formally defined an operation called vector subtraction. Our next theorem is a bit different from several of the others in the list. Rather than making a declaration (“the zero vector is unique”) it is an implication (“if…, then…”) and so can be used in proofs to convert a vector equality into two possibilities, one a scalar equlaity and the other a vector equality. It should remind you of the situation for complex numbers. If α,β and αβ = 0, then α = 0 or β = 0. This critical property is the driving force behind using a factorization to solve a polynomial equation.

Theorem SMEZV
Scalar Multiplication Equals the Zero Vector
Suppose that V is a vector space and α . If αu = 0, then either α = 0 or u = 0.

Proof   We prove this theorem by breaking up the analysis into two cases. The first seems too trivial, and it is, but the logic of the argument is still legitimate.

Case 1. Suppose α = 0. In this case our conclusion is true (the first part of the either/or is true) and we are done. That was easy.

Case 2. Suppose α0.

u = 1u  Property O = 1 ααu α0 = 1 α αu  Property SMA = 1 α 0  Hypothesis = 0  Theorem ZVSM

So in this case, the conclusion is true (the second part of the either/or is true) and we are done since the conclusion was true in each of the two cases.

Example PCVS
Properties for the Crazy Vector Space
Several of the above theorems have interesting demonstrations when applied to the crazy vector space, C (Example CVS). We are not proving anything new here, or learning anything we did not know already about C. It is just plain fun to see how these general theorems apply in a specific instance. For most of our examples, the applications are obvious or trivial, but not with C.

Suppose u C.
Then, as given by Theorem ZSSM,

0u = 0(x1,x2) = (0x1 + 0 1,0x2 + 0 1) = (1,1) = 0

And as given by Theorem ZVSM,

α0 = α(1, 1) = (α(1) + α 1,α(1) + α 1) = (α + α 1, α + α 1) = (1, 1) = 0

Finally, as given by Theorem AISM,

(1)u = (1)(x1,x2) = ((1)x1 + (1) 1,(1)x2 + (1) 1) = (x1 2, x2 2) = u

Subsection RD: Recycling Definitions

When we say that V is a vector space, we then know we have a set of objects (the “vectors”), but we also know we have been provided with two operations (“vector addition” and “scalar multiplication”) and these operations behave with these objects according to the ten properties of Definition VS. One combines two vectors and produces a vector, the other takes a scalar and a vector, producing a vector as the result. So if u1,u2,u3 V then an expression like

5u1 + 7u2 13u3

would be unambiguous in any of the vector spaces we have discussed in this section. And the resulting object would be another vector in the vector space. If you were tempted to call the above expression a linear combination, you would be right. Four of the definitions that were central to our discussions in Chapter V were stated in the context of vectors being column vectors, but were purposely kept broad enough that they could be applied in the context of any vector space. They only rely on the presence of scalars, vectors, vector addition and scalar multiplication to make sense. We will restate them shortly, unchanged, except that their titles and acronyms no longer refer to column vectors, and the hypothesis of being in a vector space has been added. Take the time now to look forward and review each one, and begin to form some connections to what we have done earlier and what we will be doing in subsequent sections and chapters. Specifically, compare the following pairs of definitions:
Definition LCCV and Definition LC
Definition SSCV and Definition SS
Definition RLDCV and Definition RLD
Definition LICV and Definition LI

Subsection READ: Reading Questions

  1. Comment on how the vector space m went from a theorem (Theorem VSPCV) to an example (Example VSCV).
  2. In the crazy vector space, C, (Example CVS) compute the linear combination
    2(3,4) + (6)(1,2).
  3. Suppose that α is a scalar and 0 is the zero vector. Why should we prove anything as obvious as α0 = 0 such as we did in Theorem ZVSM?

Subsection EXC: Exercises

M10 Define a possibly new vector space by beginning with the set and vector addition from 2 (Example VSCV) but change the definition of scalar multiplication to

αx = 0 = 0 0 α ,x 2

Prove that the first nine properties required for a vector space hold, but Property O does not hold.

This example shows us that we cannot expect to be able to derive Property O as a consequence of assuming the first nine properties. In other words, we cannot slim down our list of properties by jettisoning the last one, and still have the same collection of objects qualify as vector spaces.  
Contributed by Robert Beezer

T10 Prove each of the ten properties of Definition VS for each of the following examples of a vector space:
Example VSP
Example VSIS
Example VSF
Example VSS  
Contributed by Robert Beezer

The next three problems suggest that under the right situations we can “cancel.” In practice, these techniques should be avoided in other proofs. Prove each of the following statements.

T21 Suppose that V is a vector space, and u,v,w V . If w + u = w + v, then u = v.  
Contributed by Robert Beezer Solution [836]

T22 Suppose V is a vector space, u,v V and α is a nonzero scalar from . If αu = αv, then u = v.  
Contributed by Robert Beezer Solution [836]

T23 Suppose V is a vector space, u0 is a vector in V and α,β . If αu = βu, then α = β.  
Contributed by Robert Beezer Solution [837]

Subsection SOL: Solutions

T21 Contributed by Robert Beezer Statement [835]

u = 0 + u  Property Z = w + w + u  Property AI = w + w + u  Property AA = w + w + v  Hypothesis = w + w + v  Property AA = 0 + v  Property AI = v  Property Z

T22 Contributed by Robert Beezer Statement [835]

u = 1u  Property O = 1 ααu α0 = 1 α αu  Property SMA = 1 α αv  Hypothesis = 1 ααv  Property SMA = 1v = v  Property O

T23 Contributed by Robert Beezer Statement [835]

0 = αu + αu  Property AI = βu + αu  Hypothesis = βu + (1) αu  Theorem AISM = βu + (1)αu  Property SMA = βu + αu = β αu  Property DSA  By hypothesis, u0, so Theorem SMEZV implies 0 = β α α = β