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Section 2.1 Big Theorem
We will discover what the following have in common.
Subsection 2.1.1 Similar Problems
The following problems will illustrate properties of systems of linear equations and related matrices that are essentially the same.
Checkpoint 2.1.1 .
Determine which of the following augmented matrices (linear systems) have no, one, or multiple solutions. Explain why.
\(\displaystyle \left[ \begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -5 \\ 0 & 0 & 1 & 6 \end{array} \right] \)
\(\displaystyle \left[ \begin{array}{rrr|r} 1 & 0 & -3 & 2 \\ 0 & 1 & 6 & 11 \\ 0 & 0 & 0 & 0 \end{array} \right] \)
\(\displaystyle \left[ \begin{array}{rrr|r} 1 & 0 & -7 & 9 \\ 0 & 1 & -2 & 6 \\ 0 & 0 & 0 & 3 \end{array} \right] \)
\(\displaystyle \left[ \begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -5 \\ 1 & 1 & 1 & 3 \end{array} \right] \)
\(\displaystyle \left[ \begin{array}{rrr|r} 1 & 0 & -3 & 2 \\ 0 & 1 & 6 & 11 \\ 1 & 1 & 3 & 13 \end{array} \right] \)
\(\displaystyle \left[ \begin{array}{rrr|r} 1 & 0 & -7 & 9 \\ 0 & 1 & -2 & 6 \\ 1 & 1 & -9 & 18 \end{array} \right] \)
Checkpoint 2.1.2 .
For each of the following matrices determine which have solutions to \(A\vec{x}=\vec{b}\text{.}\) Determine which matrices have an inverse. Explain why they do or do not have an inverse.
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\left[ \begin{array}{r} 2 \\ -5 \\ 6 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 6 \\ 0 & 0 & 0 \end{array} \right]\left[ \begin{array}{r} 2 \\ 11 \\ 0 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -7 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{array} \right]\left[ \begin{array}{r} 9 \\ 6 \\ 3 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array} \right]\left[ \begin{array}{r} 2 \\ -5 \\ 3 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 6 \\ 1 & 1 & 3 \end{array} \right]\left[ \begin{array}{r} 2 \\ 11 \\ 13 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -7 \\ 0 & 1 & -2 \\ 1 & 1 & -9 \end{array} \right]\left[ \begin{array}{r} 9 \\ 6 \\ 18 \end{array} \right]\)
Checkpoint 2.1.3 .
For each of the following matrices determine whether \((0,0,0)\) is the only solution to \(A\vec{x}=\vec{0}\text{.}\) Explain why or why not.
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 6 \\ 0 & 0 & 0 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -7 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 6 \\ 1 & 1 & 3 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -7 \\ 0 & 1 & -2 \\ 1 & 1 & -9 \end{array} \right]\)
Checkpoint 2.1.4 .
For each of the following matrices determine whether \(A\vec{x}=\vec{b}\) has at least one solution for all \(\vec{b}\) (this means you can pick any \(\vec{b}\text{,}\) do not look for it below). Explain why or why not.
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 6 \\ 0 & 0 & 0 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -7 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 6 \\ 1 & 1 & 3 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -7 \\ 0 & 1 & -2 \\ 1 & 1 & -9 \end{array} \right]\)
Checkpoint 2.1.5 .
For each of the following matrices determine whether the columns span \(\R^3\) (the set of all three entry vectors). Explain why or why not.
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 6 \\ 0 & 0 & 0 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -7 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 6 \\ 1 & 1 & 3 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -7 \\ 0 & 1 & -2 \\ 1 & 1 & -9 \end{array} \right]\)
Checkpoint 2.1.6 .
For each of the following matrices determine whether the columns are linearly independent. Explain why or why not.
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 6 \\ 0 & 0 & 0 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -7 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 6 \\ 1 & 1 & 3 \end{array} \right]\)
\(\displaystyle \left[ \begin{array}{rrr} 1 & 0 & -7 \\ 0 & 1 & -2 \\ 1 & 1 & -9 \end{array} \right]\)
Checkpoint 2.1.7 .
If a matrix \(A\) has an inverse, will it be the identity matrix after row reduction?
Subsection 2.1.2 Big Theorem Begins
The following theorem will grow to encompass most of what you learn this semester.
Theorem 2.1.8 . Matrix Inversion Theorem Parts 1-8.
The following statements about an \(n \times n\) matrix \(A\) are logically equivalent.
\(A\) has a pivot position in each row.
\(A\vec{x}=\vec{b}\) 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.
\(A^{-1}\) exists.
