A set of r vectors with r > n elements are always Linearly Dependent How?

Question

Let r = No. of Vectors and n = No. of elements in each vector A set of r vectors with r > n elements are always Linearly Dependent, Can anyone please explain How?

Answer 1

Consider 

$r = no. \ of \ vector \ equations \ or  \ rows \ of \ a \ matrix$

$n= no.\ of \ variables \ in \ vector \ equations \ or \ columns \ of \ a \ matrix$

Now in your case $r>n$ . Such types of systems where number of rows or equations are greater than the number of columns or variables are known as OVER-DETERMINED system.

In general over determined system are inconsistent and have NO solutions.however there are some cases where  solution exist

• the system of equations have linearly dependent equations e.g. $y=x+1$ is linearly dependent to $2y=2x+2$ so if such equations exists then after removing linearly dependent equations the number of equations becomes less then the number of variables we have infinite solutions.
• Next if after removing linearly dependent equations the number of equations is equal to the number of number of variables we have exactly one solution.

So the only cases where the statement in your question is true , is when the system of equations (vectors) have linearly dependent equations