Exercise FS.C20 Robert Beezer

Example FSAG concludes with several questions. Perform the analysis suggested by these questions.

Exercise FS.C25 Robert Beezer

Given the matrix $A$ below, use the extended echelon form of $A$ to answer each part of this problem. In each part, find a linearly independent set of vectors, $S$, so that the span of $S$, $\spn{S}$, equals the specified set of vectors. \begin{equation*} A= \begin{bmatrix} -5 & 3 & -1 \\ -1 & 1 & 1 \\ -8 & 5 & -1 \\ 3 & -2 & 0 \end{bmatrix} \end{equation*}

  1. The row space of $A$, $\rsp{A}$.
  2. The column space of $A$, $\csp{A}$.
  3. The null space of $A$, $\nsp{A}$.
  4. The left null space of $A$, $\lns{A}$.

Solution

Exercise FS.C26 Robert Beezer

For the matrix $D$ below use the extended echelon form to find:

  1. A linearly independent set whose span is the column space of $D$.
  2. A linearly independent set whose span is the left null space of $D$.
\begin{align*} D&= \begin{bmatrix} -7 & -11 & -19 & -15\\ 6 & 10 & 18 & 14 \\ 3 & 5 & 9 & 7 \\ -1 & -2 & -4 & -3 \end{bmatrix} \end{align*}

Solution

Exercise FS.C41 Robert Beezer

The following archetypes are systems of equations. For each system, write the vector of constants as a linear combination of the vectors in the span construction for the column space provided by Theorem FS and Theorem BNS (these vectors are listed for each of these archetypes).

Archetype A, Archetype B, Archetype C, Archetype D, Archetype E, Archetype F, Archetype G, Archetype H, Archetype I, Archetype J

Exercise FS.C43 Robert Beezer

The following archetypes are either matrices or systems of equations with coefficient matrices. For each matrix, compute the extended echelon form $N$ and identify the matrices $C$ and $L$. Using Theorem FS, Theorem BNS and Theorem BRS express the null space, the row space, the column space and left null space of each coefficient matrix as a span of a linearly independent set.

Archetype A, Archetype B, Archetype C, Archetype D/Archetype E, Archetype F, Archetype G/Archetype H, Archetype I, Archetype J, Archetype K, Archetype L

Exercise FS.C60 Robert Beezer

For the matrix $B$ below, find sets of vectors whose span equals the column space of $B$ ($\csp{B}$) and which individually meet the following extra requirements.

  1. The set illustrates the definition of the column space.
  2. The set is linearly independent and the members of the set are columns of $B$.
  3. The set is linearly independent with a “nice pattern of zeros and ones” at the top of each vector.
  4. The set is linearly independent with a “nice pattern of zeros and ones” at the bottom of each vector.
\begin{equation*} B= \begin{bmatrix} 2 & 3 & 1 & 1\\ 1 & 1 & 0 & 1\\ -1 & 2 & 3 & -4 \end{bmatrix} \end{equation*}

Solution

Exercise FS.C61 Robert Beezer

Let $A$ be the matrix below, and find the indicated sets with the requested properties. \begin{equation*} A= \begin{bmatrix} 2 & -1 & 5 & -3\\ -5 & 3 & -12 & 7\\ 1 & 1 & 4 & -3 \end{bmatrix} \end{equation*}

  1. A linearly independent set $S$ so that $\csp{A}=\spn{S}$ and $S$ is composed of columns of $A$.
  2. A linearly independent set $S$ so that $\csp{A}=\spn{S}$ and the vectors in $S$ have a nice pattern of zeros and ones at the top of the vectors.
  3. A linearly independent set $S$ so that $\csp{A}=\spn{S}$ and the vectors in $S$ have a nice pattern of zeros and ones at the bottom of the vectors.
  4. A linearly independent set $S$ so that $\rsp{A}=\spn{S}$.

Solution

Exercise FS.M50 Robert Beezer

Suppose that $A$ is a nonsingular matrix. Extend the four conclusions of Theorem FS in this special case and discuss connections with previous results (such as Theorem NME4).

Exercise FS.M51 Robert Beezer

Suppose that $A$ is a singular matrix. Extend the four conclusions of Theorem FS in this special case and discuss connections with previous results (such as Theorem NME4).