EE 345 | Linear Independence

Learning Outcomes

Linear independence highlights the properties pivot columns have
\(\{ {\bf v}_1, \ldots, {\bf v}_n \}\) are linearly independent if \(V= \begin{bmatrix} {\bf v}_1 & \cdots & {\bf v}_n \end{bmatrix}\) has a pivot in every column
The set \(\{ {\bf v}_1, \ldots, {\bf v}_n \}\) of linearly independent vectors in a linear space \({\mathcal L}\) is maximal for \({\mathcal L}\) if \(\{ {\bf v}_1, \ldots, {\bf v}_n, {\bf x}\}\) becomes linearly dependent for any other \(\bf x\) from the space \(\mathcal L\)
Every vector of a linear space can be written uniquely as a linear combination of vectors from a maximally independent set of the linear space
Maximal linearly independent sets of a linear space \(=\) basis
All bases/maximal linearly independent sets in a linear space have the same size
The size of a basis/maximal linearly independent set in a linear space is a fundamental property of the linear space, called its dimension