1.4 Linear Independence

What is linear independence

• A set of vectors $({\vec{v}}_1,{\vec{v}}_2,\dots ,{\vec{v}}_k)\in {\mathbb{R}}^n$ is linearly independent if the only scalars $c_1,c_2,\dots c_k$ that satisfy the equation $c_1{\vec{v}}_1+c_2{\vec{v}}_2+\cdots +c_n{\vec{v}}_n=0$

  • is the trivial solution where all c’s = 0 otherwise the set is called linearly dependent

• EX.

  • is (1,0,0), (0,1,0), (1,1,0) LI or LD?$1\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]+1\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]+(-1)\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
    • therefore its linearly dependent
  • is (1,0,0), (0,1,0), (0,0,1) LI or LD

1 & 0 & 0 & |0 \
0 & 1 & 0 & |0 \
0 & 0 & 1 & |0
\end{bmatrix}\implies \begin{align}
c_{1}=0 \
c_{2}=0 \
c_{3}=0
\end{align}$$
- and the only solution to this is the trivial solution therefore the set is LI

• Notes!

  • when doing RREF make sure everything has no free variables
    • if it does it is LD