Linear Independence of Vectors

When they are dependent or independent #

Similar to functions, If a set of vectors are dependent, then the equation: $c_1v_1+c_2v_2+…+c_nv_n$ can have a nonzero solution $c_i\ne0$. If they’re independent, then the only solution is for all of them $c_i$ to be 0.

Theorem #

The set of vectors $\vec{v_{1}},\vec{v_{2}},…\vec{v_{n}}$ is linearly dependent iff atleast one element of the set is a linear combination of remaining other vectors.

If they’re dependent then #

$$ \begin{align*} c_1v_1+c_2v_2+…+c_nv_n=0\\ \exists i \text{ such that }c_i\ne0\\ v_{i}=\frac{1}{c_{i}}(-c_1v_1-c_2v_2…-c_nv_{n)} \end{align*} $$ ∴ $v_i$ is expressed as a linear combination of other vectors.

If one of them can be expressed as a linear combination then #

$$ \begin{align*} v_{i}=c_1v_1+c_2v_2+…+c_nv_n\\ c_1v_1+c_2v_2+…-v_i+…c_nv_n=0\\ c_{i}: \text{coefficient of }v_{i}=-1\\ c_{i}\ne 0 \end{align*} $$ ∴ the set of vectors are dependent.

Check and find dependence if possible #

We can convert the equation $$ c_1\vec{v_1}+c_2\vec{v_2}+…+c_n\vec{v_n}=0 $$

into a homogeneous system of linear equations

Let $v_k=<a_{1k},a_{2k},a_{3k},…,a_{dk}>$ (d-dimensional vectors)

[!Homogeneous System] $$ \begin{align*} a_{11}c_{1}+a_{12}c_2+…a_{1n}c_n=0\\ a_{21}c_{1}+a_{22}c_{2}+…a_{2n}c_n=0\\ .\\ .\\ .\\ a_{d1}c_{1}+a_{d2}c_{2}+…a_{dn}c_{n}=0 \end{align*} $$ (A|0) is the augmented matrix Each of the columns take up each of the ’n’ vectors.

We can utitilize Rank and solution of linear equations:

• If rank of matrix=number of vectors => only 1 solution which is all $c_i=0$ => system of vectors are independent
• If rank of matrix< number of vectors => infinite solutions => system of vectors are dependent
  • We can solve the augmented matrix to find the dependency by solving for $c_i$s.