In previous sections, we studied the general concept of vector spaces and subspaces. Now we turn our attention to three important subspaces associated with any matrix: the row space, column space, and null space. Understanding these subspaces is crucial for several reasons:
Understanding Solutions to Linear Systems: The column space tells us which vectors \(\mathbf{b}\) allow the equation \(A\mathbf{x} = \mathbf{b}\) to have a solution. The null space gives us all solutions to the homogeneous equation \(A\mathbf{x} = \mathbf{0}\text{.}\)
Matrix Rank and Dimension: The dimensions of these subspaces reveal fundamental properties of the matrix, including its rank and nullity, which measure the "information content" of the matrix.
Finding Bases Efficiently: The row space and column space provide natural ways to find bases for the spaces spanned by the rows and columns of a matrix, respectively.
Applications: These concepts are essential in data analysis, computer graphics, optimization, and many other fields where we need to understand the fundamental structure of linear transformations.
In this section, we will define these three important subspaces, explore their properties, and learn computational techniques for finding bases and dimensions.
Subsection4.3.1Definitions of Row, Column, and Null Spaces
Row, Column and Null Spaces.
Let \(A\) be a \(m\times n\) matrix, and partition \(A = \left[\begin{array}{c}
\alpha_{1}\\
\alpha_{2}\\
\vdots\\
\alpha_{m}
\end{array}\right]=[\beta_{1}\ \beta_{2}\ \ldots \ \beta_{n}]\text{.}\)
The row space of the matrix \(A\) is the set \(\{x_{1}\alpha_{1}+x_{2}\alpha_{2}+\ldots+x_{m}\alpha_{m}|x_{i}\in \mathbb{R}\}\text{,}\) denoted by \(\operatorname{Row}(A)\)
The Column space of the matrix \(A\) is the set \(\{x_{1}\beta_{1}+x_{2}\beta_{2}+\ldots+x_{n}\beta_{n}|x_{i}\in \mathbb{R}\}\text{,}\) denoted by \(\operatorname{Col}(A)\)
The dimension of the row space is \(\underline{\qquad\qquad}\text{,}\) which equals to the number of nonzero rows of the reduced echelon form of the matrix \(A\text{,}\) and which equals to the number of pivot column of the matrix \(A\text{.}\)
The dimension of the column space is \(\underline{\qquad\qquad}\text{,}\) which equals to the number of pivot column of the matrix \(A\text{,}\) and which equals to the number of nonzero rows of the reduced echelon form of the matrix \(A\text{.}\)
The dimension of the null space is \(\underline{\qquad\qquad}\text{,}\) which equals to the number of free variables of the linear system \(A\mathbf{x}=\mathbf{0}\text{.}\)
