Commit 7bb13b27 authored by Jim Hefferon's avatar Jim Hefferon

edited gr1

parent 8d66b8b4
This diff is collapsed.
......@@ -249,6 +249,35 @@
\end{ans}
\begin{ans}{One.I.1.26}
Here $S_0=\set{(1,1)}$
\begin{equation*}
\begin{linsys}{2}
x &+ &y &= &2 \\
x &- &y &= &0
\end{linsys}
\;\grstep{0\rho_2}\;
\begin{linsys}{2}
x &+ &y &= &2 \\
& &0 &= &0
\end{linsys}
\end{equation*}
while $S_1$ is a proper superset because it
contains at least two points: $(1,1)$ and~$(2,0)$.
In this example the solution set does not change.
\begin{equation*}
\begin{linsys}{2}
x &+ &y &= &2 \\
2x &+ &2y &= &4
\end{linsys}
\;\grstep{0\rho_2}\;
\begin{linsys}{2}
x &+ &y &= &2 \\
& &0 &= &0
\end{linsys}
\end{equation*}
\end{ans}
\begin{ans}{One.I.1.27}
\begin{exparts}
\partsitem Yes, by inspection the given equation results from
\( -\rho_1+\rho_2 \).
......@@ -273,7 +302,7 @@
\end{exparts}
\end{ans}
\begin{ans}{One.I.1.27}
\begin{ans}{One.I.1.28}
If \( a\neq 0 \) then the solution set of the first equation is
\( \set{(x,y)\suchthat x=(c-by)/a} \).
Taking $y=0$ gives the solution $(c/a,0)$, and since the second
......@@ -288,7 +317,7 @@
\( 0x+3y=6 \) and \( 0x+6y=12 \).
\end{ans}
\begin{ans}{One.I.1.28}
\begin{ans}{One.I.1.29}
We take three cases: that $a\neq 0$, that $a=0$ and
$c\neq 0$, and that both $a=0$ and $c=0$.
......@@ -344,7 +373,7 @@
Note that \( a=0 \) and \( c=0 \) gives that \( ad-bc=0 \).
\end{ans}
\begin{ans}{One.I.1.29}
\begin{ans}{One.I.1.30}
Recall that if a pair of lines share two distinct points then
they are the same line.
That's because two points determine a line, so these
......@@ -356,7 +385,7 @@
share at least two points (which makes them the same line).
\end{ans}
\begin{ans}{One.I.1.30}
\begin{ans}{One.I.1.31}
For the reduction operation of multiplying $\rho_i$ by a nonzero
real number $k$, we have that \( (s_1,\ldots,s_n) \) satisfies
this system
......@@ -472,7 +501,7 @@
as required.
\end{ans}
\begin{ans}{One.I.1.31}
\begin{ans}{One.I.1.32}
Yes, this one-equation system:
\begin{equation*}
0x+0y=0
......@@ -480,7 +509,7 @@
is satisfied by every \( (x,y)\in\Re^2 \).
\end{ans}
\begin{ans}{One.I.1.32}
\begin{ans}{One.I.1.33}
Yes.
This sequence of operations swaps rows \( i \) and \( j \)
\begin{equation*}
......@@ -492,7 +521,7 @@
so the row-swap operation is redundant in the presence of the other two.
\end{ans}
\begin{ans}{One.I.1.33}
\begin{ans}{One.I.1.34}
Swapping rows is reversed by swapping back.
\begin{eqnarray*}
\begin{linsys}{3}
......@@ -558,7 +587,7 @@
\end{equation*}
\end{ans}
\begin{ans}{One.I.1.34}
\begin{ans}{One.I.1.35}
Let \( p \), \( n \), and \( d \) be the number of
pennies, nickels, and dimes.
For variables that are real numbers, this system
......@@ -581,7 +610,7 @@
is the only solution using natural numbers.
\end{ans}
\begin{ans}{One.I.1.35}
\begin{ans}{One.I.1.36}
Solving the system
\begin{equation*}
\begin{linsys}{2}
......@@ -595,7 +624,7 @@
Thus the second item, 21, is the correct answer.
\end{ans}
\begin{ans}{One.I.1.36}
\begin{ans}{One.I.1.37}
\answerasgiven
A comparison of the units and hundreds columns of this
addition shows that there must be a carry from the tens column.
......@@ -615,7 +644,7 @@
was \( 47474+5272=52746 \).
\end{ans}
\begin{ans}{One.I.1.37}
\begin{ans}{One.I.1.38}
\answerasgiven
Eight commissioners voted for $B$.
To see this, we will use the given information to study how many voters
......@@ -659,7 +688,7 @@
uncommon when individual choices are pooled.
\end{ans}
\begin{ans}{One.I.1.38}
\begin{ans}{One.I.1.39}
\answerasgiven
\textit{We have not used ``dependent'' yet;
it means here that Gauss'
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......@@ -113,6 +113,8 @@ by combining upwards.
\end{equation*}
The answer is \( x=1 \), \( y=1 \), and \( z=2 \).
\end{example}
Using one entry to clear out the rest of a column is
called \definend{pivoting}\index{pivoting} on that entry.
Note that the row combination operations in the first stage proceed from column
one to column three while the combination operations in the third stage proceed
......
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......@@ -23,7 +23,7 @@ using many examples as well as extensive and careful exercises.
The developmental approach is what most recommends this book
so I will elaborate.
The courses at the beginning of a mathematics program
Courses at the beginning of a mathematics program
focus less on theory and more on calculating.
Later courses
ask for mathematical maturity:~the ability to follow different
......@@ -74,7 +74,7 @@ the second chapter starts with the definition of a real vector space.
In the schedule below, this occurs by the end of the third week.
Another example of this book's emphasis on motivation and naturalness
is that the third chapter on linear maps
is that the third chapter, on linear maps,
does not begin with the definition of homomorphism.
Rather, we start with the definition of isomorphism, which
is natural: students themselves
......
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