Linear dependence and independence. Linear operator.

Linear dependence and independence. Linear operator.


	Linear dependence and independence. Linear operator.

The concept of linear independence plays the central role in understanding what is the basis and dimension of a linear space.

Definition 1 (Linear dependence and independence) A set of vectors

$\begin{displaymath} v_1,\; v_2,\; \dots v_m \end{displaymath}$

is called linearly independent if, and only if, the system

$\begin{displaymath} v_1 \cdot x_1 + v_2 \cdot x_2 + \dots + v_m x_m = 0 \end{displaymath}$

has only trivial solution:

$\begin{displaymath} x_1=0,\; x_2=0,\; \dots x_m =0. \end{displaymath}$

Otherwise, the set is linearly dependent.

In order to establish whether the set

$\begin{displaymath} v_1,\; v_2,\; \dots v_m \end{displaymath}$

is dependent or independent one can consider the matrix

$\begin{displaymath} \left(\begin{array}{cccc} v_{11} & v_{12} & \dots & v_{1m} ... ...vdots \ v_{n1} & v_{n2} & \dots & v_{nm} \end{array}\right) \end{displaymath}$

where

$\begin{displaymath} v_1 = \left(\begin{array}{c} v_{11}\ v_{21}\ \vdots \... ... v_{1m}\ v_{2m}\ \vdots \ v_{nm} \end{array}\right). \end{displaymath}$

Then after performing Gaussian elimination procedure we obtain matrix having the following structure

$\begin{displaymath} \left(\begin{array}{cccc} 1 & \star & \dots & \star \ 0 ... ...ts & \ddots & \vdots \ 0 & 0 & \dots & 1 \end{array}\right) \end{displaymath}$

We look for non-zero leading coefficients in the rows and collect their corresponding columns from

$\begin{displaymath} v_1,\; v_2,\; \dots v_m \end{displaymath}$

Those columns build a basis in the linear space spanned by

$\begin{displaymath} v_1,\; v_2,\; \dots v_m \end{displaymath}$

This linear space is denoted by

$\begin{displaymath} spann(v_1,\; v_2,\; \dots v_m). \end{displaymath}$

The formal definition of a basis sounds as follows.

Definition 2 (Basis, dimension) The maximal number of linearly independent columns in

$\begin{displaymath} v_1,\; v_2,\; \dots v_m \end{displaymath}$

is called a basis in

$\begin{displaymath} spann(v_1,\; v_2,\; \dots v_m). \end{displaymath}$

The number of vectors in the basis is the dimension of

$\begin{displaymath} spann(v_1,\; v_2,\; \dots v_m). \end{displaymath}$

Let us consider an example.

Example 1 Find a basis for the following set of vectors.

$\begin{displaymath} \left(\begin{array}{c} 1\ 2\ 3\ 4\ 5 \end{arra... ...egin{array}{c} 0\ 2\ 4\ 6\ 8 \end{array}\right). \end{displaymath}$

After Gaussian eliminations the matrix

$\begin{displaymath} \left(\begin{array}{cccc} 1& 2 & 0 & 0 \ 2& 4 & 1 & 2 \\... ...& 2 & 4 \ 4& 8 & 3 & 6 \ 5&10 & 4 & 8 \end{array}\right) \end{displaymath}$

takes the form

$\begin{displaymath} \left(\begin{array}{cccc} 1& 2 & 0 & 0 \ 0& 0 & 1 & 2 \\... ... 0 & 0 \ 0& 0 & 0 & 0 \ 0& 0 & 0 & 0 \end{array}\right). \end{displaymath}$

Looking at the columns containing non-zero leading coefficients we conclude that

$\begin{displaymath} v_1,\; v_3 \end{displaymath}$

is a basis for

$spann(v_1,\; v_2,\; v_3, \; v_4).$

Now consider a linear mapping

$\begin{displaymath} A: \;{\rm R}^n \; \to \; {\rm R}^m. \end{displaymath}$

Definition 3 (Linear operator) A mapping

is called a linear operator if, and only if,

$\begin{displaymath} A(x+y)=Ax + Ay \mbox{ and } A(\lambda \cdot x) = \lambda \cdot Ax \end{displaymath}$

After fixing bases in

${\rm R}^n$ ${\rm R}^m$

one can assign a matrix to a linear operator. Hence, one can identify the linear operator

$\begin{displaymath} A: \;{\rm R}^n \; \to \; {\rm R}^m. \end{displaymath}$

with its matrix

$\begin{displaymath} A=\left( \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1... ...ts\ a_{m1} & a_{m2} & \dots & a_{mn}\ \end{array}\right). \end{displaymath}$

There are two important linear spaces associated with a linear operator A.

Image of A,

$\begin{displaymath} im(A)=\{y\in {\rm R}^m; \mbox{ such that there exists } x\in {\rm R}^n \mbox{ and }\;\; y=Ax \} \end{displaymath}$

Kernel (zero space or null space) of A

$\begin{displaymath} ker(A)=\{x\in {\rm R}^n; \mbox{ such that there exists } Ax=0 \} \end{displaymath}$

The dimension of

is called rank of A and denoted as

$\begin{displaymath} rank(A). \end{displaymath}$

Next:ExercisesUp:Linear dependence and independence.Previous:Linear dependence and independence.

Sergey Nikitin 2005-01-31