## Linear Independence

\[S\]

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.