##### Linearly independent solutions inLinear Algebra

The number of linearly independent solutions of 'm' homogeneous linear equations in 'n' variables ..AX=0 is (n-r) where r is the rank of a matrix..could somebody explain me by giving proper reasoning as how?...Thanks

''The number of linearly independent solutions of 'm' homogeneous linear equations in 'n' variables ..AX=0 is (n-r) where r is the rank of a matrix.''

It means that when we are given m equations and n variables , then after applying row transformations and converting it into row- echelon form we get a matrix where r is the rank of matrix , these r equations are actually linearly independent solutions , while n-r are linearly dependent solutions which can be eliminated.

Example: \( \begin{bmatrix}3&2&4\\5&10&7\\6&4&8\end{bmatrix}\)

\( \begin{bmatrix}3&2&4\\5&10&7\\0&0&0\end{bmatrix}\)

here , rank of matrix is 2 // no. of non-zero rows or equations

and n-r =3-2 = 1 is no. of linearly independent solution.