Column Space, Row Space, NULL Space and Orthogonal Space of a Matrix

1. The Columns/Vectors of any \(M \times N\) Matrix \(A\) form/belong to the Column Space of that Matrix. Also the Product of Matrix \(A\) with any \(N \times 1\) Vector/Matrix \(X\) gives a \(M \times 1\) Vector/Matrix \(Y\) that belongs to the Column Space of the Matrix \(A\). That is if
   
   \(AX=Y\)   ...(1)
   
   then \(Y\) belongs to Column Space of Matrix \(A\).
   
   Conversely if any \(M \times 1\) Vector/Matrix \(Y\) belongs to Column Space of an \(M \times N\) Matrix \(A\), then there exists an \(N\times 1\) Vector/Matrix \(X\) which when Multiplied by Matrix \(A\) gives Vector/Matrix \(Y\). The Vector \(X\) can be found out by Solving System of Linear Equations given by equation (1).
   
   Also, if the Vector/Matrix \(X\) Does Not Exist for a given Matrix \(A\) and Vector/Matrix \(Y\) (i.e the System of Linear Equation given by equation (1) gives No Solutions), then the Vector/Matrix \(Y\) Does Not Belong to Column Space of Matrix \(A\).
2. The Rows/Co-Vectors of any \(M \times N\) Matrix \(A\) form/belong to the Row Space of that Matrix. Also the Product of Transpose of Matrix \(A\) (i.e \(A^T\)) with any \(M \times 1\) Vector/Matrix \(X\) gives a \(N \times 1\) Vector/Matrix \(Y\) whose Transpose belongs to the Row Space of the Matrix \(A\). That is, Row Space of any Matrix \(A\) is Same as Column Space of \(A^T\).
3. NULL Space of Columns of Any \(M \times N\) Matrix \(A\) is a List of All Possible \(N \times 1\) Vectors/Matrices \(X\), each of which when Multiplied by the Matrix \(A\) gives a \(M \times 1\) NULL Vector as given in the following Matrix Equation
   
   \(AX=0\)   ...(2)
   
   Given Matrix \(A\) having elements \(a_{ij}\) as following
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix}\)
   
   The NULL Space of Columns for Matrix \(A\) can be found out by Solving the System of Homogeneous Linear Equations as given below
   
   \(AX=0 \Rightarrow \begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}\)   ...(3)
   
   
4. The Orthogonal Space of Any Matrix is Same as the NULL Space of Rows of that Matrix (i.e. NULL Space of Columns of Transpose of that Matrix). Orthogonal Space of Any \(M \times N\) Matrix \(A\) is a List of All Possible \(M \times 1\) Vectors/Matrices \(X\), each of which when Multiplied by the Transpose of Matrix \(A\) gives a \(N \times 1\) NULL Vector. That is if
   
   \(A^TX=0\)   ...(4)
   
   then \(X\) belongs to Orthogonal Space of Matrix \(A\).
5. If a Vector \(X\) belongs to Orthogonal Space Matrix \(A\), then it is also Orthogonal to Any of the Vectors that Belong to Column Space of the Matrix \(A\).
6. NULL / Orthogonal Space of Columns / Rows of Any \(N \times N\) Non-Singular Square Matrix \(A\) (i.e. Determinant \(|A|\neq0\)) is an \(N\times 1\) NULL Vector.