Vector Space of a Matrix and Rank of a Matrix

1. Any \(M \times N\) Matrix (where \(M \geq 2\) and \(N \geq 1\)) consists of \(N\) Vectors/Columns with each Vector having \(M\) Components. Any such \(M \times N\) Matrix can Represent/Span a Vector Space whose Dimension can range from \(1\) to \(M\) or \(N\) (whichever is less).
   
   The Dimension of the Vector Space Represented/Spanned by such a Matrix is called the Rank of the Matrix.
2. The Rank of Any \(M \times N\) Matrix (where \(M > 1\) and \(N > 1\)) is given by the Count of Linearly Independent Vectors present in the Matrix which 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.