Linear Independence

A set of vectors  
\[S\]
  is linearly indpendent of none of the vectors in  
\[S\]
  can be expressed in terms of the others. Any set  
\[S\]
  of elements (as long as  
\[S\]
  contains more than one element) may or may not be linearly indpendent.
One way to determine independence to to write the elements of  
\[S\]
  as column vectors with respect to some basis. If the resulting matrix is square (
\[n\]
  vectors in a space of dimension  
\[n\]
 ) then the vectors are linearly independent if the determinmant of the matrix is not zero.
Example:  
\[S = \{1,1+x,1+x+x^2 \}\]
  With respect to the standard basis  
\[B= \{1,x,x^2 \}\]
 . We can write the elements of  
\[S\]
  as  
\[\begin{pmatrix}1\\0\\0\end{pmatrix}, \begin{pmatrix}1\\1\\0\end{pmatrix}, \begin{pmatrix}1\\1\\1\end{pmatrix}\]
.
The resulting matrix is
\[ \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array} \right) \]
  with determinant 1 and the elements of  
\[S\]
  are linearly independent. If  
\[S = \{1+x,1+x^2,2+x+x^2 \}\]
  With respect to the standard basis  
\[B= \{1,x,x^2 \}\]
  then we can write the elements of  
\[S\]
  as  
\[\begin{pmatrix}1\\1\\0\end{pmatrix}, \begin{pmatrix}1\\0\\1\end{pmatrix}, \begin{pmatrix}2\\1\\1\end{pmatrix}\]
.
The resulting matrix is
\[ \left( \begin{array}{ccc} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{array} \right) \]
  with determinant 0 and the elements of S are linearly dependent.
It should be noted that linear dependence or independence is a property of the set and not the basis. Chaning the basis does not affect linear dependence.
Example:  
\[S = \{1+\mathbf{i},1, 1-\mathbf{i}\}\]
  With respect to the standard basis  
\[B= \{1, \mathbf{i} \}\]
  we can write the elements of  
\[S\]
  as  
\[\begin{pmatrix}1\\1\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}1\\-1\end{pmatrix}\]
.
The resulting matrix is
\[ \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & -1 \end{array} \right) \]
.
This is not a square matrix so has no determinant. The dimesion of the set of complex numbers is 2, and there are three elemtns in this set. The order of  
\[S\]
  is greater than the dimension of the space so  
\[S\]
  is linearly dependent.

Add comment

Security code
Refresh