Linear Dependence/Independence of Vectors in a Matrix and Rank of a Matrix

1. Any \(M \times N\) Matrix (i.e a Matrix having \(M\) Rows and \(N\) Columns where \(M > 1\) and \(N > 1\)) Contains \(N\) Vectors each having \(M\) Components. The Linear Dependency Relation between All Vectors of such a Matrix is given by the Components of Vectors of the NULL Space of Columns of the Matrix. If the NULL Space of Columns of the Matrix consists of only the NULL Vector then the Vectors of the Matrix are Linearly Independent. Otherwise they are Linearly Dependent. Additionally, for the Matrix
   1. If the Number of Rows is Lesser than Number of Columns in the Matrix (i.e. \(M < N\), Number of Vectors are more than Number of Components in each Vector), then the Vectors of the Matrix are always Linearly Dependent.
   2. If the Number of Rows is Greater than or Equal to the Number of Columns in the Matrix (i.e. \(M>=N\), Number of Components in each Vector is Greater than or Equal to Number of Vectors), then the Vectors of the Matrix are Linearly Independent only if Elementry Row Operations on Matrix can convert it into a Matrix containing only Mutually Orthogonal Identitity Vectors (Implying that its NULL Space of Columns consists of only NULL Vector). Otherwize the Vectors are Linearly Dependent.
2. The Vectors/Columns in any \(N \times N\) Square Matrix are Linearly Independent only if Determinant of the Matrix is Non-Zero. Otherwise they are Linearly Dependent.
3. The Vectors/Columns in any \(M \times N\) Square or Non Square Matrix \(A\) are Linearly Independent only if Determinant of the Matrix \(A^TA\) (and/or \(A^{\dagger}A\) if \(A\) is a Complex Matrix) is Non-Zero. Otherwise they are Linearly Dependent.
4. The Count of Linearly Independent Vectors present in any \(M \times N\) Matrix (where \(M > 1\) and \(N > 1\)) is equal to the Number of Non Zero Rows in the Matrix after performing Elementary Row Operations (or Number of Non Zero Columns in the Matrix after performing Elementary Column Operations) for finding the NULL Space of the Matrix. This Count is called the Rank of the Matrix.