Column Vectors

A column vector of size \(\,n\,\) over a field \(\,K\,\) is a sequence of \(\,n\,\) scalars, written vertically:

\[\begin{split}\boldsymbol{x}\ =\ [x_{i}]_n\ \ =\ \ \left[\begin{array}{c} x_{1} \\ x_{2} \\ \ldots \\ x_{n} \end{array}\right] \,, \quad \text{where}\quad x_{i}\in K,\quad i=1,2,\ldots,n\,; \quad n\in N.\end{split}\]

The set of all such vectors is denoted by \(\,K^n.\ \) When dealing with practical problems, \(\\\) \(\ K\ \) is usually the field \(\,R\ \) of real numbers or the field \(\ C\ \) of complex numbers.

The column vectors in \(\,K^n\,\) may be added: 1

\[\begin{split}\left[\begin{array}{c} x_1 \\ x_2 \\ \ldots \\ x_n \end{array}\right] \ +\ \left[\begin{array}{c} y_1 \\ y_2 \\ \ldots \\ y_n \end{array}\right] \ :\,=\ \, \left[\begin{array}{c} x_1+y_1 \\ x_2+y_2 \\ \ldots \\ x_n+y_n \end{array}\right]\,,\end{split}\]

and multiplied by scalars \(\, c \in K\):

\[\begin{split}c \ \ \left[\begin{array}{c} x_1 \\ x_2 \\ \ldots \\ x_n \end{array}\right] \ :\,=\ \, \left[\begin{array}{c} c\; x_1 \\ c\; x_2 \\ \ldots \\ c\; x_n \end{array}\right]\,.\end{split}\]

In Sage it is equally simple:

Press Evaluate to see the outcome of the code. You can modify the code yourself too. See what happens if the vectors are of different size. In section Vectors and Matrices in Sage we explain how to indicate the underlying field.

The addition and scalar multiplication of column vectors in \(\,K^n\,\) consist in addition and multiplication of scalars in \(\,K.\ \) On that basis, it’s easy to validate the following properties, which lead to the conclusion that \(\,K^n\,\) is an abelian group under addition of column vectors.

0. \(\,\) The addition of column vectors is an internal operation in \(\,K^n.\)

1. \(\,\) The addition is associative and commutative:

   \[ \begin{align}\begin{aligned}(\boldsymbol{x} + \boldsymbol{y}) \, + \, \boldsymbol{z} \ \; = \ \; \boldsymbol{x} \, + \,(\boldsymbol{y} + \boldsymbol{z})\,,\\\boldsymbol{x}\,+\,\boldsymbol{y}\ =\ \boldsymbol{y}\,+\,\boldsymbol{x}, \qquad\forall\ \ \boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\,\in\,K^n.\end{aligned}\end{align} \]

2. \(\,\) The neutral element for addition is the zero column vector \(\ \,\boldsymbol{\theta}\ =\ \left[\begin{array}{c} 0 \\ 0 \\ \ldots \\ 0 \end{array}\right]\,.\)

3. \(\,\) For any \(\ \boldsymbol{x}\in K^n,\ \) there exists the opposite vector \(\ (-\,\boldsymbol{x})\ \) such that \(\ \boldsymbol{x} + (-\,\boldsymbol{x}) = \boldsymbol{\theta}.\)

   
   

   \(\,\) Namely, \(\,\) for \(\ \,\boldsymbol{x}\,=\, \left[\begin{array}{c} x_{1} \\ x_{2} \\ \ldots \\ x_{n} \end{array}\right]\ \,\) the opposite is \(\ \,(-\,\boldsymbol{x})\,=\, \left[\begin{array}{c} -x_{1} \\ -x_{2} \\ \ldots \\ -x_{n} \end{array}\right]\,.\)

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Furthermore, since the scalar multiplication of column vectors is distributive, both over addition of scalars and over addition of vectors:

\[(a + b)\ \boldsymbol{x}\ =\ a\,\boldsymbol{x}\ +\ b\,\boldsymbol{x}\,, \qquad a\,(\boldsymbol{x} + \boldsymbol{y})\ =\ a\,\boldsymbol{x}\,+\,a\,\boldsymbol{y}\,,\]

and satisfies the compatibility conditions

\[a\,(b\,\boldsymbol{x})\ =\ (ab)\,\boldsymbol{x},\qquad 1\,\boldsymbol{x}\ =\ \boldsymbol{x},\qquad\quad \forall\ \ a,b\in K,\quad\forall\ \ \boldsymbol{x},\boldsymbol{y}\in\ K^n,\]

the set \(\ K^n\ \) is a vector space over the field \(\,K\,\) under addition and scalar multiplication of column vectors.

Column vectors over the real field \(\,R\,\) of size two or three can be associated with geometric vectors in a plane or in the space, respectively. Namely, if \(\ \,\vec{e}_1,\,\vec{e}_2\ \,\) or \(\ \,\vec{e}_1,\,\vec{e}_2\,,\vec{e}_3\ \,\) form a basis of mutually perpendicular unit vectors, then:

\(\quad\boldsymbol{x}\ =\ \left[\begin{array}{c} x_1 \\ x_2 \end{array}\right] \quad\simeq\quad \vec{v}\ =\ x_1\,\vec{e}_1 + x_2\,\vec{e}_2\,;\)

\(\quad\boldsymbol{x}\ =\ \left[\begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array}\right] \quad\simeq\quad \vec{v}\ =\ x_1\,\vec{e}_1 + x_2\,\vec{e}_2 + x_3\,\vec{e}_3\,.\)

The relation \(\ \simeq\ \) between the column and geometric vectors has the property that

if \(\quad\boldsymbol{x}\ \simeq\ \vec{v}\quad\text{and}\quad \boldsymbol{y}\ \simeq\ \vec{w},\qquad\)

then \(\qquad\ \boldsymbol{x}+\boldsymbol{y}\ \,\simeq\ \,\vec{v}+\vec{w}\qquad\) and \(\qquad c\ \boldsymbol{x}\ \simeq\ c\ \vec{v}\ \) for all \(\ c\in R.\)

A bijective correspondence between two vector spaces over the same field (geometric vectors also form a real vector space), which preserves space operations in the above sense, is called isomorphism, and the pertinent vector spaces are said to be isomorphic.

The notion of isomorphism generalized to other algebraic structures (e.g. to algebras) will be discussed in subsequent parts of this textbook.

Experiment with Sage:

For given values \(\ x_1,\,x_2\ \) you will get the geometric image \(\ \vec{v}\ \) of the vector \(\ \,\boldsymbol{x} = \left[\begin{array}{c} x_1 \\ x_2 \end{array}\right]\).

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The symbol \(\ \ " :\,= "\ \ \) means \(\ \) “equal by definition”.