library(tidyverse)
library(dasc2594)18 Rank
Definition 18.1 Given an \(m \times n\) matrix \(\mathbf{A}\), the row space, row(\(\mathbf{A}\)) is the set of linearly independent rows of \(\mathbf{A}\). Thus, the row space is the span of the rows of \(\mathbf{A}\).
Note: the row space of \(\mathbf{A}\) is the column space of the transposed matrix \(\mathcal{A}'\).
\[ \begin{aligned} row(\mathbf{A}) = col(\mathbf{A}') \end{aligned} \]
Example 18.1 in class
Example 18.2 Find basis for row space, column space, and null space of \(\mathbf{A}\)
18.1 Rank
Definition 18.2 The rank of a matrix \(\mathbf{A}\), rank(\(\mathbf{A}\)) is the dimension of the column space col(\(\mathbf{A}\))
Theorem 18.1 (The Rank Theorem) Let \(\mathbf{A}\) be an \(m \times n\) matrix. Then the dimension of are equal. The rank of \(\mathbf{A}\) equals the number of pivot columns of \(\mathbf{A}\) and
\[ \begin{aligned} rank (\mathbf{A}) + dim(null(\mathbf{A})) = n \end{aligned} \]
Example 18.3 in class
\(\mathbf{A}\) is an \(m \times n\) matrix with dim(null(\(\mathbf{A}\))) = p. What is rank(\(\mathbf{A}\))
Example 18.4 \(\mathbf{A}\) is a 6x9 matrix. Is it possible for null(\(\mathbf{A}\)) = 2?
Theorem 18.2 (Invertible Matrix Theorm + Rank) This is an extension of the prior statement of the invertible matrix theorem Theorem 9.5 Let \(\mathbf{A}\) be an \(n \times n\) matrix. Then the following statements are equivalent (i.e., they are all either simultaneously true or false).
13) The columns of \(\mathbf{A}\) form a basis of \(\mathcal{R}^n\)
14) col(\(\mathbf{A}\)) = \(\mathcal{R}^n\)
15) dim(col(\(\mathbf{A}\))) = \(n\)
16) rank(\(\mathbf{A}\)) = \(n\)
17) null(\(\mathbf{A}\)) = \(\{\mathbf{0}\}\)
18) dim(null(\(\mathbf{A}\))) = 0