Question

A collection of vectors \(v_1,...,v_n\) is said to be linearly independent if

\[c_1v_1 + \cdots c_nv_n = 0\] \[\iff\] \[c_1=\cdots=c_n=0\]

In other words, any linear combination of the vectors that results in a zero vector is trivial.

Another interpretation of this is that no vector in the set may be expressed as a linear combination of the others. In this sense, linear independence is an expression of non-redundancy in a set of vectors.

Fact: Any linearly independent set of \(n\) vectors spans an \(n\)-dimensional space. (I.e. the collection of all possible linear combinations is \(\mathbb{R}^n\).) Such a set of vectors is said to be a basis of \(\mathbb{R}^n\). Another term for basis is minimal spanning set.

A LOT!!

• If \(A\) is an \(m\times n\) matrix and \(m>n\), if all \(m\) rows are linearly independent, then the system is overdetermined and inconsistent. The system cannot be solved exactly. This is the usual case in data analysis, and why least squares is so important.
• If \(A\) is an \(m\times n\) matrix and \(m<n\), if all \(m\) rows are linearly independent, then the system is underdetermined and there are infinite solutions.
• If \(A\) is an \(m\times n\) matrix and some of its rows are linearly dependent, then the system is reducible. We can get rid of some equations.
• If \(A\) is a square matrix and its rows are linearly independent, the system has a unique solution. (\(A\) is invertible.)

Linear algebra has a whole lot more to tell us about linear systems, so we’ll review a few basics.