A more general definition of vectors will be discussed in Part 3.
Vectors
Important properties of linear systems can be described with concept and notation of vectors. As a motivating example, let us consider a system of three equations
The term vector appears in a variety of mathematical and engineering contexts, which we will discuss in Part3 (Vector Spaces). There is no universal notation for vectors because of diversity of their applications. Until then, vector will mean an ordered set (list) of numbers. Generally speaking, concept of vector may include infinite sequence of numbers or other objects. However, in this part of tutorial, we consider only finite lists. There are many ways to represent vectors including columns, rows, or just ntuples as elements of Cartesian product of real numbers:
Remember that the form of vector representation as columns, rows, or ntuples (parentheses and comma notation) depends on you. However, you must be consistent and use the same notation for addition or scalar multiplication. You cannot add a column vector and a row vector:
Giusto Bellavitis  Michail Ostrogradsky  William Hamilton 
The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. The Italian mathematician, senator, and municipal councilor Giusto Bellavitis (18031880) abstracted the basic idea in 1835. The idea of an ndimensional Euclidean space for n > 3 appeared in a work on the divergence theorem by the Russian mathematician Michail Ostrogradsky (18011862) in 1836, in the geometrical tracts of Hermann Grassmann (18091877) in the early 1840s, and in a brief paper of Arthur Cayley (18211895) in 1846. Unfortunately, the first two authors were virtually ignored in their lifetimes. In particular, the work of Grassmann was quite philosophical and extremely difficult to read. The term vector was introduced by the Irish mathematician, astronomer, and mathematical physicist William Rowan Hamilton (18051865) as part of a quaternion.
Vectors can be described also algebraically. Historically, the first vectors were Euclidean vectors that can be expanded through standard basic vectors that are used as coordinates. Then any vector can be uniquely represented by a sequence of scalars called coordinates or components. The set of such ordered ntuples is denoted by \( \mathbb{R}^n . \) When scalars are complex numbers, the set of ordered ntuples of complex numbers is denoted by \( \mathbb{C}^n . \) Motivated by these two approaches, we present the general definition of vectors.
Rewriting vectors u and v in column form, we have
 Compute u −v and 3u −2v > \[ {\bf u} = \begin{bmatrix} 2 \\ \phantom{}1 \end{bmatrix} , \quad {\bf v} = \begin{bmatrix} 3 \\ 2 \end{bmatrix} \qquad\mbox{and} \qquad {\bf u} = \begin{bmatrix} \phantom{}1 \\ 1 \end{bmatrix} , \quad {\bf v} = \begin{bmatrix} \phantom{}4 \\ 3 \end{bmatrix} . \]

Write system of equations that is equivalent to the given vector equation.

\[ x_1 \begin{bmatrix} \phantom{}3 \\ 4 \end{bmatrix} + x_2 \begin{bmatrix} 1 \\ 2 \end{bmatrix} + x_3 \begin{bmatrix} \phantom{}5 \\ 2 \end{bmatrix} = \begin{bmatrix} \phantom{}1 \\ 1 \end{bmatrix}; \]

\[ x_1 \begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix} + x_2 \begin{bmatrix} 1 \\ \phantom{}3 \\ \phantom{}5 \end{bmatrix} + x_3 \begin{bmatrix} 2 \\ \phantom{}7 \\ \phantom{}2 \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \\ 3 \end{bmatrix}; \]

\[ x_1 \begin{bmatrix} 2 \\ 1 \\ 7 \end{bmatrix} + x_2 \begin{bmatrix} \phantom{}3 \\ 2 \\ 5 \end{bmatrix} + x_3 \begin{bmatrix} \phantom{}4 \\ 6 \\ \phantom{}1 \end{bmatrix} = \begin{bmatrix} 5 \\ 4 \\ \phantom{}2 \end{bmatrix} . \]

 Given \( \displaystyle {\bf u} = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} , \ {\bf v} = \begin{bmatrix} \phantom{}3 \\ 1 \\ 5 \end{bmatrix} , \quad\mbox{and} \quad {\bf b} = \begin{bmatrix} 5 \\ 5 \\ h \end{bmatrix} . \) For what value of h is b a linear combination of vectors u and v?
 Given \( \displaystyle {\bf u} = \begin{bmatrix} 4 \\ 3 \\ 1 \end{bmatrix} , \ {\bf v} = \begin{bmatrix} \phantom{}2 \\ 2 \\ 3 \end{bmatrix} , \quad\mbox{and} \quad {\bf b} = \begin{bmatrix} 8 \\ h \\ 9 \end{bmatrix} . \) For what value of h is b a linear combination of vectors u and v?

Rewrite the system of equations in a vector form
\[ \begin{split} 2x_1  3 x_2 + 7 x_3 &= 1 , \\ 5 x_1 2 x_2  3 x_3 &= 2 , \\ 3x_1 + 2 x_2 + 4 x_3 &= 3 . \end{split} \]
 Let \( \displaystyle {\bf u} = \begin{bmatrix} 3 \\ 1 \end{bmatrix} , \ {\bf v} = \begin{bmatrix} \phantom{}2 \\ 2 \end{bmatrix} , \quad\mbox{and} \quad {\bf b} = \begin{bmatrix} h \\ k \end{bmatrix} . \) Show that the linear equation x₁u + x₂v = b has a solution for any values of h and k.

Mark each statement True or False.
 Another notation for the vector (1, 2) is \( \displaystyle \begin{bmatrix} 1 \\ 2 \end{bmatrix} . \)
 An example of a linear combination of vectors u and v is 2v.
 Any list of of six complex numbers is a vector in ℂ^{6}.
 The vector 2v results when a vector v + u is added to the vector v − u.
 The solution set of the linear system whose augmented matrix is [ a_{1} a_{2} a_{3} b ] is the same as the solution set of the vector equation x₁a₁ + x₂a₂ + x₃a₃ = b.
 Vector addition
 Tea
 Milk