Four Fundamental Subspaces
In this article, we discuss about the four fundamental subspaces related to a matrix. They are - Column space - Row Space - Null Space -
Column Space \(C(A)\)
Let a matrix \(A\) consists of columns \(u_1,u_2,\dots u_n\). Then \(C(A) = \text{span}(u_1,u_2,u_3\dots u_n)\)
Solving \(Ax=b\) using \(C(A)\)
\(A(x)=b\) is solvable if and only if \(b\in C(A)\)
Let us look at an example to see how we use the \(C(A)\) to solve \(Ax=b\)
Is \(Ax=b\) solvable? No, because \(C(A)=\text{span}(C_1, C_2)\) and \(C_3 = C_1+C_2\) whereas the variables are only \(3\). Hence there are \(4\) equations and \(3\) unknowns and \(C(A)\) is a 2-dimensional subspace of \(\R^4.\)
Row space
The row space of a matrix \(A\) is defined as column space of \(A^T\), i.e., \(R(A)=C(A^T)\)
Note
Column Rank \(=\) Row Rank
Null Space
Null space of a matrix \(A\) is defined as \(N(A)=\{x|Ax=0\}\)
Remember
If \(A\) is invertible matrix, then \(N(A) = 0\) and \(C(A)\) is the entire space. It so happens that \(Ax=b\) has a unique solution \(x=A^{-1}b\)
Tip
We can also use the Gaussian elimination to find the null space of a matrix \(A.\)
Rank Nullity Theorem
In the gaussian elimination, the number of pivot columns is called the Rank of the matrix, i.e., Rank \(=\) Dim \((C(A)).\)
And the number of free variables is known as Nulity of the matrix, i.e., Nullity \(=\) Dim \((N(A)).\)
Now according to the theorem, if \(A\) has \(n\) columns, then $$ n=\text{Rank} + \text{Nullity} $$
Left Null Space
The Left Null Space of a matrix \(A\) is defined as the null space of \(A^T.\)
Tip
Row Space \(+\) Left Null Space \(=\) Row Space