The concept of "vector" provides a mathematical formalism that allows us to handle, with precision and rigor, the quantities that occur in natural phenomena and make predictions. A vector is the mathematical gadget that physicists use to describe the state of a quantum or mechanical system. The advent of computer algebra systems and computer numerical laboratories has precipitated a common need among engineers and scientists to learn the language of vectors and matrices, which is used heavily in applications.

 

Fields

The heart of definition of vectors is in two operations---one internal operation between vectors, usually called additions, and one outer operation of multiplication of vectors by a scalar. Of course, these two operations must satisfy some axioms or postulates.

A field is a set 𝔽 with at least two elements together with a function 𝔽 × 𝔽 ⇾ 𝔽 called addition, denoted (a, b) → a + b, and a function 𝔽 × 𝔽 ⇾ 𝔽 called multiplication, denoted (a, b) → ab, which satisfy the following axioms:
  1. Commutativity of addition and multiplication: a + b = b + a, and a ⋅ b = b ⋅ a.
  2. Associativity of addition and multiplication: a + (b + c) = (a + b) + c, and a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c.
  3. Additive and multiplicative identity: there exist two different elements 0 and 1 in 𝔽 such that a + 0 = a and a ⋅ 1 = a.
  4. Additive inverses: for every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + (−a) = 0.
  5. Multiplicative inverses: for every a ≠ 0 in F, there exists an element in 𝔽, denoted by a−1 or 1/a, called the multiplicative inverse of a, such that a ⋅ a−1 = 1.
  6. Distributivity of multiplication over addition: a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c).
Example 1: Our main field is ℚ, the set of rational numbers, that is, numbers that are represented as the quotients or ratios of integers p/q, where p, q ∈ ℤ = { 0, ±1, ±2, ±3, … }. Humans and computers can operate only with rational numbers. However, the field ℚ is not complete, which means that a limit of rational numbers is not always a rational number. Such numbers are called irrational numbers; for instance, the golden ratio (1+√5)/2 or π are irrational numbers---they cannot be expressed as ratios of integers. The set of all rational numbers and all irrational numbers is denoted by ℝ and it is called the field of real numbers.

We also need a field of complex numbers ℂ that include all ordered pairs of real numbers (x, y), x, y ∈ ℝ, equipped with two operations: addition

\[ \left( x_1 , y_1 \right) + \left( x_2 , y_2 \right) = \left( x_1 + x_2 , y_1 + y_2 \right) \]
and multiplication
\[ \left( x_1 , y_1 \right) \times \left( x_2 , y_2 \right) = \left( x_1 \, x_2 - y_1 \, y_2 , x_1 y_2 + x_2 y_1 \right) . \]
Mathematica confirms:
ComplexExpand[(x1 + I y1) (x2 + I y2)]
x1 x2 - y1 y2 + I (x2 y1 + x1 y2)
You may want to check that all field axioms are satisfied. These ordered pairs are convenient to denote as
\[ \left( x , y \right) = x + {\bf j}\,y = x + y\,{\bf j} , \qquad {\bf j}^2 = -1 . \]
Mathematicians (following L. Euler) denote the imaginary unit vector j by i. However, in physics and engineering, notation j is the usual custom.

As a less familiar example, we consider the set ℚ(√3) that consists of all numbers of the form 𝑎 + b√3, where 𝑎 and b are rational numbers. It is easy to show that addition operation satisfies all postulates of the field. Their multiplication is also closed:

\[ \left( x_1 + y_1 \sqrt{3} \right) \times \left( x_2 + y_2 \sqrt{3} \right)= x_1 x_2 + 3\,y_1 y_2 + \left( x_1 y_2 + x_2 y_1 \right) \sqrt{3} . \]
{x1 + y1*Sqrt[3]}*{x2 + y2*Sqrt[3]} // Expand
{x1 x2 + Sqrt[3] x2 y1 + Sqrt[3] x1 y2 + 3 y1 y2}
Although there are know many classes of fields, we will use only three fields: ℚ, ℝ, and ℂ. These three fields will be united in one notation 𝔽, which represents all of these.

Vector Spaces

Giuseppe Peano
Now we present axiomatic definition of vector space, which requires an explanation of this step. The first question that comes to your mind is why do practical people in science and engineering need this abstract involment if we know that all finite dimensional vector spaces are actually reduced to ℝn or ℂn? Why it is not enough to consider numerical n-tuples?

To answer this question, we first remind you that Mathematica considers vectors as lists having entries not just numerical values but alowing incorporation of other objects. This is typical in computer science to treat vectors as one-dimensional arrays. Abstract definition is necessary for further generalizations, including differential operators, tensors, exterior algebra, differentiable manifolds. and infinite dimensional vector spaces, just to mention some of them.

The first axiomatic definition of a linear space was provided by the Italian mathematician Giuseppe Peano (1858--1932) in 1888 when he published Calcolo geometrico secondo I'Ausdehnungslehre de H. Grassmann preceduto dalle operazioni della logica deduttiva. Peano credits the work of Leibniz, Mobius, Grassmann, and Hamilton as having provided him with the ideas that led to his formal calculus. In this remarkable book, Peano introduces what subsequently took a long time to become standard notation for basic set theory.

A vector space V over the field of scalars 𝔽 (either ℚ or ℝ or ℂ) is a set of elements or objects, called vectors, together with two operations that satisfy the eight axioms (or rules) listed below.
1. The first operation is an inner operation that assigns to any two vectors x and y a third vector that is commonly written as x + y and called the sum of these two vectors.
2. The second operation, is an outer operation that assigns to any scalar k∈𝔽 and vector x∈V another vector, denoted by kx∈V.

  1. Associativity of addition: \( ({\bf v} + {\bf u}) + {\bf w} = {\bf v} + ({\bf u} + {\bf w}) \) for u, v, and w ∈ V.
  2. Commutativity of addition: u + v = v + u for u, v ∈ V.
  3. Identity element of addition: there exists an element \( {\bf 0} \in V , \) called the zero vector, such that \( {\bf v} +{\bf 0} = {\bf v} \) for every vector v from V.
  4. Inverse elements of addition: for every vector v, there exists an element \( -{\bf v} \in V , \) called the additive inverse of v, such that \( {\bf v} + (-{\bf v}) = {\bf 0} . \)
  5. Compatibility of scalar multiplication with field multiplication: \( a(b{\bf v}) = (ab){\bf v} \) for any scalars a and b and arbitrary vector v.
  6. Identity element of scalar multiplication: \( 1{\bf v} = {\bf v} , \) where 1 denotes the multiplicative identity.
  7. Distributivity of scalar multiplication with respect to vector addition: \( k\left( {\bf v} + {\bf u}\right) = k{\bf v} + k{\bf u} \) for any scalar k and arbitrary vectors v and u.
  8. Distributivity of scalar multiplication with respect to field addition: \( \left( a+b \right) {\bf v} = a\,{\bf v} + b\,{\bf v} \) for any two scalars a and b and arbitrary vector v. ■

A vector space must have at least one element, its zero vector. Thus, a one-element vector space is the smallest possible.

A one-element vector space containing the zero vector is a trivial space.
Example 2: The set ℝn or ℂn of all ordered n-tuples of real or complex numbers is our first familiar example of vector spaces. This space has a standard basis: \( {\bf e}_1 = (1,0,0,\ldots ,0 ) ,\quad {\bf e}_2 = (0,1,0,\ldots , 0 ), \ldots , {\bf e}_n = (0,0,\ldots , 0,1) .\) In ℝ³ these unit vectors are denoted by
\[ {\bf i} = \left( 1, 0 ,0 \right) , \qquad {\bf j} = \left( 0,1,0 \right) , \qquad {\bf k} = (0,0,1) . \]
Therefore, every vector \( {\bf v} = \left( v_1 , v_2 , v_3 \right) \) can be expressed as a linear combination of standard unit vectors:
\[ {\bf v} = \left( v_1 , v_2 , v_3 \right) = v_1 {\bf i} + v_2 {\bf j} + v_3 {\bf k} . \]
We shall check just two of the conditions, the closure conditions.

Indeed, the result of the vector sum

\[ \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix} \]
is a column array with two real entries, and so is a member of the plane ℝ² (we assume for simplicity that 𝔽 is the field of real numbers). Condition for multiplication by a scalar is verified similarly:
\[ k \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} = \begin{pmatrix} k\,x_1 \\ k\,y_1 \end{pmatrix} . \]
■

 

Vector Notations

Mathematics is a language that seeks to describe Nature in a form that is as purely logical as can be imagined. As part of that effort, indeed the alphabet of the language of mathematics, a notational system has emerged to present ideas in shorthand. This is a human convenience often specific to the setting, not the essence of what is being achieved. It is nice to say "v = { e₁, e₂, e₃ }" rather than "the vector, v, is composed of a set of three ordered and numbered elements, each with the name 'e.'" However, it is important to remember that what matters is the physical reality of what "e" represents. The symbol "e" may represent an element of Nature such as plutonium or other entry in the Periodic Table. Alternatively, it may represent an electrical charge. These are very different physical things. One must always focus on what a symbol stands for rather than the symbol itself.

The definition of vectors summarizes the most important properties that are used to model real life phenomena that humans have observed for thousands of years. The name "vector" can be applied to completely different physical objects and their mathematical counterparts. For example, a force in quantum mechanics requires different approach from a similar physical description in classical mechanics.

There is no universal description of vectors: in one area of science vectors are used differently from similar objects in another discipline. It has become customary to use lowercase letters in boldface type for vector notation (as, for instance, x or v). However, this notation is not enough for applications of vectors and more detailed description is needed. As you will learn in near future, all (finite dimensional) vector spaces look very similar to Cartesian products of real numbers or complex numbers, discussed in Example 2.

As a brief reminder, a Cartesian product of two sets A and B consists of all ordered pairs (𝑎, b), where 𝑎 ∈A and b ∈ B,

\[ A \times B = \left\{ (a, b) \ : \ a \in A, \quad b\in B \right\} . \]
The Cartesian product require no algebraic structure for neither sets A or B. If sets A and B carry some algebraic structure (e.g., they are vector spaces over the same field), then we can define a suitable structure on the product set as well. So, direct product is like Cartesian product, but with some additional structure. For example, if A and B are vector spaces, their direct product forms a vector space with respect to element-wise addition
\[ (a_1 , b_1 ) + \left( a_2 , b_2 \right) = \left( a_1 + a_2 , b_1 + b_2 \right) \]
and scalar multiplication
\[ k \left( a , b \right) = \left( k\,a , k\,b \right) , \qquad k \in \mathbb{F}. \]
Obviously, the definition of direct product can be extended to arbitrary finite number of vector spaces. For example, we can define a product of n real number fields:
\[ \mathbb{R}^n = \mathbb{R} \times \mathbb{R} \times \cdots \times \mathbb{R} . \]
Elements of these products consist of ordered sets (known as lists or arrays) of numbers:
\[ {\bf v} = \left( v_1 , v_2 , \ldots , v_n \right) \in \mathbb{F}^n , \qquad v_i \in \mathbb{F}, \quad i = 1,2,\ldots , n. \]

This representation v = (v1, v2, ... ,vn) of vectors as n-tuples is called the comma-delimited form. However, since a vector in 𝔽n is just a list of its n components in a specific order, any notation that displays those components in the correct order is a valid way of representing the vector. For example, the vector can be written as a column vector
\[ {\bf v} = \left( \begin{array}{c} v_1 \\ v_2 \\ \vdots \\ v_m \end{array} \right) \qquad \mbox{also written as } \qquad {\bf v} = \left[ \begin{array}{c} v_1 \\ v_2 \\ \vdots \\ v_m \end{array} \right] . \]
The set of all column vectors of length n is denoted (in this tutorial) as 𝔽n,1 because they are n×1 matrices. In quantum physics these vectors are also called ket-vectors and are denoted by |v>. On the other hand, a row vector is called the bra-vector and it is denoted as
\[ <{\bf v}\,| = \left[ v_1 , v_2 , \ldots , v_n \right] \in \mathbb{F}^{1,n}. \]
Here entries \( v_i \) are known as the components of the bra-vector <v|. The set of all row vectors is denoted by 𝔽1,n, the set of 1×n matrices over field 𝔽. These three notations (as n-tuple, column vector, or row vector) are most popular in applications, but this does not mean that they are the only possible alternatives. One may represent vectors by diagonal matrices (Mathematica has a special command in this case, DiagonalMatrix)
\[ \Lambda = \begin{bmatrix} a_1 & 0 & 0 & \cdots & 0 \\ 0 & a_2 & 0 & \cdots & 0 \\ 0 & 0 & a_3 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0&0&0& \cdots & a_n \end{bmatrix} . \]
Now you see that there are many options for vector descriptions and any restriction in their notation is not wise because of diversity of vector applications. The comma-delimited form is most appropriate when vectors are considered by themselves used only given vector operations (addition and scalar multiplication). However, when vectors are transferred by matrix multiplications, then column notation is preferable because A x is a column vector for any matrix A and column vector x. Similarly, when matrix A operates on vector x from right to left, we use row (or bra) notation because x A is a row vector. In these two cases of matrix multiplications, diagonal matrices are not suitable as A Λ is not a diagonal matrix.

Since all European languages write from left to right, we also apply operators (including matrices) at the left of the vector as A x (where A is a matrix and x is a vector). Therefore, we mostly will write vectors in a column form

These three notations (as n-tuple, column vector, or row vector) can be used, but you may not mix them. For example adding a column vector and a row vector is not permitted because they are matrices of different dimensions. The same is true for n-tuples and either of bra- or ket-vectors.

 

How Mathematica handles vectors

A string is a grouping or ordering collection of characters or symbols with quotes around them such as s = "mary had a little lamb". Operations on strings are necessarily different from operations on numbers.
s = "mary had a little lamb"
mary had a little lamb
Mathematica "sees" an object based on its "Head" which is Part[0] of the expression
Part[s, 0]
String
s[[0]]
String
Head[s]
String
StringLength[s]
22
Spaces are counted just like any other character
Characters[s]
{"m", "a", "r", "y", " ", "h", "a", "d", " ", "a", " ", "l", "i", \ "t", "t", "l", "e", " ", "l", "a", "m", "b"}
Length[%]
22
You can convert letters into numbers by assigning each letter its corresponding number in the alphabet. The letter "m" is the first and twenty-first character in s.
StringPosition[s, "m"][[All, 2]]
{1, 21}
The letter "m" appears as characters 1 and 21; and the letter "a" appears as 2, 7, 10 and 20.
Flatten[StringPosition[s, #] & /@ {"m", "a"}, 1][[All, 2]] (*InputForm[Alphabet[]]*)
{1, 21, 2, 7, 10, 20}
You can reconstruct the original expression from the position of each of its letters (although I am not sure why anyone would want to do that)
StringPosition[InputForm[Alphabet[]], #] & /@ s
% == s
mary had a little lamb
True

Mathematica does not distinguish columns from rows, so the user should specify which object is in use. This will be familiar to Physics students who are not taught that there is a distinction. But it will be vexatious to the Finance and Economics students who have spent years learning that columns and rows must be treated separately. This demonstrates the "human and application-specific" nature of notation convention. Part of the enormous benefit of Mathematica and evident genius of Stephen Wolfram is to to be as general as possible at all times. Logic is best served by simplicity and generality. Mathematica expresses symbolic logic extremely well, given the constraints of computer hardware (compare an American English keyboard to the keyboard used in a different country). Disposing of the "row-column" convention and substituting for it instead the "every list is a vector and every vector is a list" paradigm at the operational level is a feat to be admired, not an impediment to understanding.

In Mathematica, there are no sets since Mathematica requires that an ordering be placed on its data, and so it deals with lists that can be treated as multisets if you ignore the order of the elements in the lists. The list of elements a, b, and c is embraced into curly brackets as L= { a, b, c } which Mathematica considers as a row-vector. The empty list is { }, and L[[k]] is the k-th element of list L.

Let us define two vectors, one as a row denoted v1, and another as a column denoted v2:

v1 = { 1, 2, 3}
v2 = {{1}, {2}, {3}}
We can also check that they are row vector and column vector, respectively, with the command:
v1 // TraditionalForm
v2 // MatrixForm
v1 is a 1×3 matrix; while v2 would be column vector, which is a 3×1 matrix. Therefore, these two vectors have different dimensions and cannot be added. Nevertheless, Mathematica does not care and if one enter
v1 + v2
the output will be the 3-element column vector:
\( \displaystyle \begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix} \) { {2}, {4}, {6} }

If you wish all matrices to appear in MatrixForm by default, this code sets Mathematica to render accordingly, permitting you to visually distinguish between columns and rows.

$Post := If[MatrixQ[#1], MatrixForm[#1], #1] &
Operations on lists are unaffected by the $Post setting
2*v1
v1. v1
{2, 4, 6}
14

When you want to add row vector v1 with column vector v2 by entering v1 + v2, Mathematica assumes you want to add v1 as if it were a column...

{{1}, {2}, {3}} + v2
\( \displaystyle \begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix} \)
rather than assuming you want to add v2 as a row
v1 + Flatten[v2]
{2, 4, 6}


Vectors in Mathematica are built, manipulated and accessed similarly to matrices (see next section). However, as simple lists (“one-dimensional,” not “two-dimensional” such as matrices that look more tabular), they are easier to construct and manipulate. They will be enclosed in brackets ( [,] ) which allows us to distinguish a vector from a matrix with just one row, if we look carefully. The number of “slots” in a vector is not referred to in Mathematica as rows or columns, but rather by “size.”

In Mathematica, defining vectors and matrices is done by typing every row in curly brackets:

v ={1,2^6 ,Sin[x]}
Out[1]= {1, 2^6, Sin[x]}
v is a vector with three components, v[[1]] =1, v[[2]]= 26, and v[[3]]=Sin[x]. Double bracket notation is abbreviation for the Mathematica command Part:
v[[2]]
Part[v, 2]
64
An element can be a number or a function or even a vector or other entity such as a data or an image. We usually denote vectors with lower case bold letters while matrices with upper case bold letters. Some textbooks use the convention of placing a vector-like "hat" or arrow over the symbol to indicate it is a vector, thus:
\[ \hat{\bf v} \qquad \mbox{or} \qquad \vec{\bf v} . \]
But this is just another human convenience which does not change the substance or meaning.

A column vector can be constructed from curly brackets shown here { }. A comma delineates each row. The output, however, may not look like a column vector. To fix this you must either invoke the $Post command above as we have done here or individually call //MatrixForm

u={1,2,3,4}
MatrixForm[u]
[5]//MatrixForm
1
2
3
4

Constructing a row vector is very similar to constructing a column vector, except two sets of curly brackets are used. Again, the $Post setting puts the row vector in the familiar format

n={{1,2,3,4}}
MatrixForm[n]
[7]//MatrixForm
{ 1 2 3 4 }
To illustrate again how Mathematica usually does not distinguish column-vectors from row-vectors, let us define two vectors:
a = {1, 0, 2};
a // TraditionalForm
Out[4]= \( \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} \)
b = {{1, 0, 2}}
b // MatrixForm
Out[6]= (1, 0, 2)

Both a vector and a matrix can be multiplied by a scalar; with the operation being *. Matrices and vectors can be added or subtracted only when their dimensions are the same.

 

Example 3: Let us consider a set ℭ[0, 1] of continuous functions on interval [0, 1]. Recall that |a,b| denotes an open or closed or semiclosed interval on the real axis ℝ . The set ℭ(|𝑎, b|) of all continuous functions on the interval |𝑎, b| is a vector space. Indeed, for two functions f and g in ℭ([0, 1], ℝ) and c ∈ ℝ, we make the obvious definitions of f + g and c f:
\[ ( f + g)(x) := f (x) + g(x), \qquad (c\, f )(x) := c\, f (x) . \]
The 0 is the function that is identically zero. We leave it as an exercise to check that all the vector space conditions are satisfied.

The space ℭ¹([0, 1], ℝ) of continuously differentiable functions is a subspace of ℭ([0, 1], ℝ).


Example 4: For any nonnegative integer n, a polynomial of degree n is an expression of the form
\[ p_n (x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n , \]
where each coefficient ai is a scalar/number. The zero polynomial is the polynomial having all coefficients to be zero. Two polynomials \( p_n (x) = a_0 + a_1 x + \cdots + a_n x^n \) and \( q_n (x) = b_0 + b_1 x + \cdots + b_m x^m \) , where for simplicity \( n\ge m , \) can be added:
\[ p_n (x) + q_m (x) = \left( a_0 + b_0 \right) + \left( a_1 + b_1 \right) x + \cdots + \left( a_m + b_m \right) x^m + a_{m+1} x^{m+1} + \cdots + a_n x^n , \]
and multiplied by a constant
\[ k\, p_n (x) = k\,a_0 + k\,a_1 x + k\,a_2 x^2 + \cdots + k\,a_n x^n . \]
Under these operations of addition and scalar multiplication, the set 𝒫≤n of all polynomials of degree not exceeding n is a vector space. Moreover, there is a one-to-one correspondence between 𝒫n and ℝn+1:
\[ a_0 + a_1 x + a+_2 x^2 + \cdots + a_n x^n \quad \mapsto \quad \left[ a_0 , a_ 1 , a_2 , \ldots , a_n \right] \in \mathbb{R}^{n+1} . \]
In the above, the variable x is really just a formal placeholder. By setting x equal to a real or complex number, we obtain a function. So the space 𝒫≤n[x] can be thought of as a subspace of ℭ(ℝ, ℝ). If we restrict the range of x to [0, 1], 𝒫≤n[x] can be identified with a subspace of ℭ(ℝ, ℝ.
Example 5: Let ℓ ∞(ℕ) or simply ℓ ∞ consist of objects of the form
\[ {\bf u} = \left[ u_0 , u_1 , u_2 , \ldots , u_n , \ldots \right] , \]
in which u0, u1, u3, … is an infinite sequence of real numbers (or complex numbers, if you need) that do not exceed some fixed number. We define two infinite sequences to be equal if their corresponding components are equal. The inner operation of addition is defined as
\begin{align*} {\bf u} + {\bf v} &= \left[ u_0 , u_1 , u_2 , \ldots , u_n , \ldots \right] + \left[ v_0 , v_1 , v_2 , \ldots , v_n , \ldots \right] \\ &= \left[ u_0 + v_0 , u_1 + v_1 , u_2 + v_2 , \ldots , u_n + v_n , \ldots \right] . \end{align*}
Correspondingly, the scalar multiplication is defined as
\[ k\,{\bf u} = k \left[ u_0 , u_1 , u_2 , \ldots , u_n , \ldots \right] = \left[ k\,u_0 , k\,u_1 , k\,u_2 , \ldots , k\,u_n , \ldots \right] . \]
It can be shown that the set ℓ∞(ℕ) is a vector space over the field of real numbers (or complex numbers, if you wish).

One can define a similar vector space of all doubly infinite sequences of numbers, denoted as ℓ∞(ℤ), usually written in a row form rather than column

\[ {\bf u} = \left[ \cdots , u_{-n} , \ldots , u_0 , u_1 , u_2 , \ldots , u_n , \ldots \right] . \]
It is also assumed that all entries in each sequence are absolutely bounded.
Example 6: Let V = ℚ² and define addition and multiplication by a rational number k∈ℚ as follows. If u = [u₁, u₂] and v = [v₁, v₂], then their sum is
\[ {\bf u} + {\bf v} = \left[ u_{1} + v_1 , u_2 + v_2 \right] \]
and
\[ k\,{\bf u} = \left[ 0, k\,u_2 \right] . \]
This addition operation is the standard one from ℚ², but the scalar multiplication is not. The last axiom fails to hold for certain vectors. For example, if u = [u₁, u₂], with u₁ ≠ 0, then
\[ 1\,{\bf u} = \left[ 0, 1\,u_2 \right] = \left[ 0, u_2 \right] \ne {\bf u} . \]
Thus, V is not a vector space with the stated operations.
Example 7: Let V be the set of positive numbers, let u = u and v = v be any vectors (i.e., positive real numbers) in V, and let k be any scalar. Define the operations on V to be
\begin{align*} u + v &= u\cdot v \qquad \mbox{(vector addition is numerical multiplication)}, \\ k\,u &= u^k \qquad \mbox{(scalar multiplication is numerical exponentiation)}. \end{align*}
Thus, for example, 1 + 1 = 1 and (2)(1) = 1² = 1. Note that 1 plays the role of zero element. Nevertheless, the set pf positive numbers with these two operations satisfies all eight axioms for being vector space.
  1. Let ℚ(√2) be the set of all numbers of the form 𝑎 + b√2, where 𝑎 and b are rational numbers. Is this set a field?
  2. Let ℚ(√2, √3) be the set of all numbers of the form 𝑎 + b√2 + c√3, where 𝑎 b, and c are rational numbers. Is this set a field?
  3. Let F = ℝ² be the set of all ordered pairs (𝑎, b) of real numbers, 𝑎, b ∈ ℝ, equipped with regular addition \[ (a_1 , b_1 ) + (a_2 , b_2 ) = \left( a_1 + a_2 , b_1 + b_2 \right) \] and multiplication \[ \left( a_1 , b_1 \right) \times \left( a_2 , b_2 \right) = \left( a_1 a_2 , b_1 b_2 \right) . \] Is this set F a field ?
  4. Let F = ℝ² be the set of all ordered pairs (𝑎, b) of real numbers, 𝑎, b ∈ ℝ, equipped with regular addition \[ (a_1 , b_1 ) + (a_2 , b_2 ) = \left( a_1 + a_2 , b_1 + b_2 \right) \] and multiplication \[ \left( a_1 , b_1 \right) \times \left( a_2 , b_2 \right) = \left( a_1 a_2 + b_1 b_2 , a_1 b_2 + a_2 b_1 \right) . \] Is this set F a field ?
  5. Name the zero vector for each of these vector spaces.
    1. The space of degree three polynomials under the natural operations.
    2. The space of 3×4 matrices.
    3. The space { f : [0..1] → ℝ | f is continuous }.
  6. Let V be the set of all ordered pairs of real numbers and consider the following addition and scalar multiplication operations on u = (u₁, u₂) and v = (v₁, v₂):
    \[ {\bf u} + {\bf v} = (u_1 + v_1 +1, u_2 + v_2 -1), \qquad k\, {\bf u} = (k\,u_1 , k\,u_2 ) . \]
    1. Compute u + v and ku for u = (0, 3), v = (, −4), and k = 3.
    2. Show that (0, 0) ≠ 0.
    3. Show that (−1, 1) = 0.
    4. Show that Axiom 4 holds by producing an ordered pair −u such that u + (−u) = 0 for u = (u₁, u₂).
    5. Find two vector space axioms that fail to hold.
  7. Find the additive inverse, in the vector space, of the vector.
    1. In vector space of polynomials of degree at most 3, ℙ₃, the vector −5 −3x + 2x².
    2. In the space of square 2×2 matrices,
      \[ \begin{bmatrix} \phantom{-}3 & 0 \\ -2 & 5 \end{bmatrix} . \]
    3. In { \( \displaystyle a\,e^x + b\,e^{−x}\, |\, a, b \in \mathbb{R} \) }, the space of functions of the real variable x under the natural operations, the vector \( \displaystyle 5\,e^x - 3\,e^{−x} . \)
  8. In the following exercises, determine whether each set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail.
    1. The set of 2×2 matrices with real entries where the (2,1) entry is zero, under the usual matrix operations.
    2. The set of all pairs of real numbers that have the form [x, y], where x ≥ 0, with the standard operations on ℝ².
    3. The set of all n-tuples of real numbers that have the form [x, x, … , x], with the standard operations on ℝn.
    4. The set of rational numbers ℚ under the field of scalars ℝ.
    5. The set of all triples of real numbers with the standard vector addition but with scalar multiplication defined by
      \[ k \left[ x, y, z \right] = \left[ k^2 x , k^2 y , k^2 z \right] . \]
    6. The set
      \[ \left\{ \left[ x, y, z, w \right]^{\mathrm{T}} \,: \, x -y + z - w = 0 \right\} \]
      under the operations inherited from ℝ4.
    7. The set of all triples of real numbers under the operations inherited from ℝ3 that satisfy
      \[ \left\{ \left[ x, y, z \right]^{\mathrm{T}} \,: \, x^2 + y^2 + z^2 = 1 \right\} . \]
    8. The subset of the vector space ℭ[−1, 1] of real-valued continuous functions that vanish at the origin.
    9. The set of 2×2 matrices under the usual matrix operations,
      \[ \left\{ \begin{bmatrix} a & 1 \\ b & c \end{bmatrix} \, : \, a,b,c \in \mathbb{R}_{+} \right\} , \]
      where ℝ+ is the set of all nonnegative real numbers.
    10. The set of polynomials of degree up to two with nonnegative real coefficients.
  9. Show that the set of continuous functions on interval [0, 1] such that \( \displaystyle \int_0^1 f(x)\,{\text d} x = 0 \) is a vector space.

 

  1. Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
  2. Axler, S., Linear A;gebra done Right, Third edition, Springer, 2015, doi: 10.1007/978-3-319-11080-6
  3. Beezer, R.A., A First Course in Linear Algebra, 2017.
  4. Fitzpatrick, S., Linear Algebra: A second course, featuring proofs and Python. 2023.