The Linear Combination Lemma

Decide if the matrices are row equivalent.

• (a)  $\begin{pmatrix}1&2\\ 4&8\end{pmatrix},\begin{pmatrix}0&1\\ 1&2\end{pmatrix}$     (b)  $\begin{pmatrix}1&0&2\\ 3&-1&1\\ 5&-1&5\end{pmatrix},\begin{pmatrix}1&0&2\\ 0&2&10\\ 2&0&4\end{pmatrix}$     (c)  $\begin{pmatrix}2&1&-1\\ 1&1&0\\ 4&3&-1\end{pmatrix},\begin{pmatrix}1&0&2\\ 0&2&10\\ \end{pmatrix}$     (d)  $\begin{pmatrix}1&1&1\\ -1&2&2\end{pmatrix},\begin{pmatrix}0&3&-1\\ 2&2&5\end{pmatrix}$     (e)  $\begin{pmatrix}1&1&1\\ 0&0&3\end{pmatrix},\begin{pmatrix}0&1&2\\ 1&-1&1\end{pmatrix}$

Which of these matrices are row equivalent to each other?

• (a)  $\begin{pmatrix}1&3\\ 2&4\end{pmatrix}$     (b)  $\begin{pmatrix}1&5\\ 2&10\end{pmatrix}$     (c)  $\begin{pmatrix}1&-1\\ 3&0\end{pmatrix}$     (d)  $\begin{pmatrix}2&6\\ 4&10\end{pmatrix}$     (e)  $\begin{pmatrix}0&1\\ -1&0\end{pmatrix}$     (f)  $\begin{pmatrix}3&3\\ 2&2\end{pmatrix}$

Produce three other matrices row equivalent to the given one.

• (a)  $\begin{pmatrix}1&3\\ 4&-1\end{pmatrix}$     (b)  $\begin{pmatrix}0&1&2\\ 1&1&1\\ 2&3&4\end{pmatrix}$

Perform Gauss’s Method on this matrix. Express each row of the final matrix as a linear combination of the rows of the starting matrix.

Describe the matrices in each of the classes represented in Example 2.10.

Describe all matrices in the row equivalence class of these.

• (a)  $\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$     (b)  $\begin{pmatrix}1&2\\ 2&4\end{pmatrix}$     (c)  $\begin{pmatrix}1&1\\ 1&3\end{pmatrix}$

How many row equivalence classes are there?

Can row equivalence classes contain different-sized matrices?

How big are the row equivalence classes?

• Show that for any matrix of all zeros, the class is finite.

• Do any other classes contain only finitely many members?

Give two reduced echelon form matrices that have their leading entries in the same columns, but that are not row equivalent.

Show that any two $n\!\times\!n$ nonsingular matrices are row equivalent. Are any two singular matrices row equivalent?

Describe all of the row equivalence classes containing these.

• (a)  $2\!\times\!2$ matrices     (b)  $2\!\times\!3$ matrices     (c)  $3\!\times\!2$ matrices     (d)  $3\!\times\!3$ matrices

• Show that a vector $\vec{\beta}_{0}$ is a linear combination of members of the set $\{\hskip 1.00006pt\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\hskip 1.00006pt\}$ if and only if there is a linear relationship $\vec{0}=c_{0}\vec{\beta}_{0}+\cdots+c_{n}\vec{\beta}_{n}$ where $c_{0}$ is not zero. (Hint. Watch out for the $\vec{\beta}_{0}=\vec{0}$ case.)

• Use that to simplify the proof of Lemma 2.5.

Three truck drivers went into a roadside cafe. One truck driver purchased four sandwiches, a cup of coffee, and ten doughnuts for $$8.45$. Another driver purchased three sandwiches, a cup of coffee, and seven doughnuts for $$6.30$. What did the third truck driver pay for a sandwich, a cup of coffee, and a doughnut?

The Linear Combination Lemma says which equations can be gotten from Gaussian reduction of a given linear system.

1. Produce an equation not implied by this system.

2. Can any equation be derived from an inconsistent system?

Extend the definition of row equivalence to linear systems. Under your definition, do equivalent systems have the same solution set?

In this matrix

the first and second columns add to the third.

• Show that remains true under any row operation.

• Make a conjecture.

• Prove that it holds.