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--- lecture_1 [2015/01/21 10:26] – created rupert
+++ lecture_1 [2016/01/22 19:13] (current) – old revision restored (2015/01/22 12:02) rupert
@@ Line 1: / Line 1: @@
-====== Chapter 1: Linear systems of equations ======
+====== Chapter 1: Systems of linear equations ======
 ===== Linear equations =====
+==== First example: a linear equation in two variables ====
 Consider the equation \[ 2x+5y=7.\]
-This is an equation in two //[[variable]]s//, or //[[indeterminate]]s//, $x$ and $y$.
+This is an equation in two //[[variables]]//, or //[[indeterminates]]//, $x$ and $y$.
-A [[solution]] of this equation will be a pair of numbers $(a,b)\in \mathbb{R}^2$ so that if we replace $x$ with $a$ and replace $y$ with $b$, then the equation becomes true.
+A [[solution]] of this equation is a pair of numbers $(a,b)\in \mathbb{R}^2$ so that if we replace $x$ with $a$ and replace $y$ with $b$, then the equation becomes true.
 In other words, so that $2a+5b$ really is equal to $7$.
@@ Line 12: / Line 14: @@
   * $(3,1)$ is not a solution, because $2\times 3+5\times 1\ne 7$
   * $(1,1)$ is a solution, because $2\times 1+5\times 1=7$
-  * Other solutions include $(0,\tfrac 75)$, $(0.5,1.4)$, $(6,-1)$, $(-\tfrac32,2)$, ...
+  * Other solutions include $(0,\tfrac 75)$, $(0.5,1.2)$, $(6,-1)$, $(3.5,0)$, $(-\tfrac32,2)$, ...
+We can't make a complete list of all solutions, since there are //infinitely many// solutions in $\mathbb{R}^2$. However, we can draw the set of all solutions as a subset of $\mathbb{R}^2$. This turns out to be a straight line:
+{{ ::g1.jpg-000.jpg?nolink&600 |}}
+We say that the equation $2x+5y=7$ is a [[linear equation in two variables]].
+==== Definition ====
+{{page>linear equation in two variables}}
+==== More examples of linear equations in two variables ====
-We can't make a complete list of all solutions, since there are //infinitely many// solutions in $\mathbb{R}^2$. However, we can draw the set of all solutions. This turns out to be a line:
+  * $y-x=1$ {{ ::s-000.jpg?nolink&300 |}}
+  * $x-y=0$ {{ ::r-000.jpg?nolink&300 |}}
+  * $x=0\iff 1x+0y=0$ {{ ::r-001.jpg?nolink&300 |}}