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Section 4.5 Rank
Goals.
We will consider various properties of dimensions of a row space including
which matrices have the same row spaces,
the comparative size of row and column spaces, and
the comparative size of row and null spaces.
Subsection 4.5.1 Row Spaces
Activity 4.5.1 .
In this activity we will see how to write a simple and useful description of a given row space.
\begin{equation*}
A=\left[ \begin{array}{*{4}{r}}
1 & 1 & 1 & 7 \\
1 & 1 & 2 & 0 \\
1 & 1 & 1 & 21 \\
-3 & -3 & -1 & -35 \\
\end{array} \right],
\end{equation*}
\begin{equation*}
B=\left[ \begin{array}{*{4}{r}}
1 & 1 & 1 & 2 \\
2 & 2 & 3 & 4 \\
3 & 3 & 2 & 7 \\
0 & 0 & 2 & -2 \\
\end{array} \right].
\end{equation*}
\begin{equation*}
C=\left[ \begin{array}{*{4}{r}}
1 & -2 & -4 & 4 \\
2 & -3 & -5 & 11 \\
3 & -5 & -9 & 16 \\
2 & -2 & -2 & 15 \\
\end{array} \right].
\end{equation*}
(a) Find bases for the row spaces of \(A\text{,}\) \(B\text{,}\) and \(C\) by row reducing the matrices. Use technology.
(b) Compare the dimensions of these bases.
(c) Compare the contents of these row spaces (do they share any vectors)?
(d) If two matrices can be row reduced to the same matrix, what will be true about their row spaces?
(e) If a matrix can be row reduced to the identity matrix, what is its row space?
Subsection 4.5.2 Column Spaces
Activity 4.5.2 .
In this activity we notice which columns we can use for a basis of the column space.
\begin{equation*}
A=\left[ \begin{array}{*{4}{r}}
1 & 1 & 1 & 7 \\
1 & 1 & 2 & 0 \\
1 & 1 & 1 & 21 \\
-3 & -3 & -1 & -35 \\
\end{array} \right],
\end{equation*}
\begin{equation*}
B=\left[ \begin{array}{*{4}{r}}
1 & 1 & 1 & 2 \\
2 & 2 & 3 & 4 \\
3 & 3 & 2 & 7 \\
0 & 0 & 2 & -2 \\
\end{array} \right].
\end{equation*}
\begin{equation*}
C=\left[ \begin{array}{*{4}{r}}
1 & -2 & -4 & 4 \\
2 & -3 & -5 & 11 \\
3 & -5 & -9 & 16 \\
2 & -2 & -2 & 15 \\
\end{array} \right].
\end{equation*}
(a) Find bases for the column spaces of \(A\text{,}\) \(B\) and \(C\) by column reducing. This is the same as row reducing the transpose which you can do with software.
Keep track of where each column of your basis was originally in the matrix.
(b) Row reduce \(A\text{,}\) \(B\) and \(C\) using technology. Which columns have a pivot position?
(c) Compare the locations of the columns with pivot positions to the columns that formed a basis from the column reduction.
Activity 4.5.3 .
In this activity we compare the size of the row and column spaces for a given matrix.
\begin{equation*}
A=\left[ \begin{array}{*{4}{r}}
1 & 1 & 1 & 7 \\
1 & 1 & 2 & 0 \\
1 & 1 & 1 & 21 \\
-3 & -3 & -1 & -35 \\
\end{array} \right],
\end{equation*}
\begin{equation*}
B=\left[ \begin{array}{*{4}{r}}
1 & 1 & 1 & 2 \\
2 & 2 & 3 & 4 \\
3 & 3 & 2 & 7 \\
0 & 0 & 2 & -2 \\
\end{array} \right].
\end{equation*}
\begin{equation*}
C=\left[ \begin{array}{*{4}{r}}
1 & -2 & -4 & 4 \\
2 & -3 & -5 & 11 \\
3 & -5 & -9 & 16 \\
2 & -2 & -2 & 15 \\
\end{array} \right].
\end{equation*}
(a) Find bases for the column spaces of \(A\text{,}\) \(B\) and \(C\text{.}\)
(b) Compare the dimension of the column spaces of \(A\) and \(B\) to the dimension of their row spaces.
(c) What can you conclude about the dimensions of the row and column spaces for any matrix? Note the result of
Activity 4.5.2 provides a connection.
Subsection 4.5.3 Null Spaces
Activity 4.5.4 .
In this activity we compare the dimensions of row and null spaces for a matrix.
\begin{equation*}
A=\left[ \begin{array}{*{4}{r}}
1 & 1 & 1 & 7 \\
1 & 1 & 2 & 0 \\
1 & 1 & 1 & 21 \\
-3 & -3 & -1 & -35 \\
\end{array} \right],
\end{equation*}
\begin{equation*}
B=\left[ \begin{array}{*{4}{r}}
1 & 1 & 1 & 2 \\
2 & 2 & 3 & 4 \\
3 & 3 & 2 & 7 \\
0 & 0 & 2 & -2 \\
\end{array} \right].
\end{equation*}
\begin{equation*}
C=\left[ \begin{array}{*{4}{r}}
1 & -2 & -4 & 4 \\
2 & -3 & -5 & 11 \\
3 & -5 & -9 & 16 \\
2 & -2 & -2 & 15 \\
\end{array} \right].
\end{equation*}
(a) Find bases for the null spaces of \(A\text{,}\) \(B\text{,}\) and \(C\) using technology.
(b) What is the maximum dimension for a row space for \(4 \times 4\) matrices? Call this \(n\text{.}\)
(c) Compare the dimensions of the row and null spaces with \(n\text{.}\)
(d) What can you conclude about the dimension of the row space and the dimension of the null space?
Subsection 4.5.4 Terminology
Definition 4.5.1 . Rank.
The dimension of the row space of a matrix is called the rank of the matrix.
Theorem 4.5.2 . Rank and Null.
\(\mbox{rank}A+\mbox{dim Nul} A = n\) for an \(n \times n\) matrix.
We should consider what these facts add to our big theorem.
Theorem 4.5.3 . Matrix Inversion Theorem Parts 1-14.
The following statements about an \(n \times n\) matrix \(A\) are logically equivalent.
\(A^{-1}\) exists.
\(A\) has a pivot position in each row.
\(A\) has a pivot position in each column.
\(A\vec{x}=\vec{b}\) has exactly one solution for every \(\vec{b}\text{.}\)
\(A\vec{x}=\vec{0}\) has only the trivial solution.
The columns of \(A\) span \(\R^n\text{.}\)
The rows of \(A\) span \(\R^n\text{.}\)
The columns of \(A\) are independent.
The rows of \(A\) are independent.
The transformation defined by \(A\) is one-to-one.
The transformation defined by \(A\) is onto.
\(\det(A)\) is non-zero.
\(A\) is full rank.
\(\mbox{Nul}(A)=\{\vec{0}\}\text{.}\)
