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--- lecture_6_slides [2017/02/06 18:16] – rupert
+++ lecture_6_slides [2017/02/08 17:34] (current) – rupert
@@ Line 2: / Line 2: @@
 ===== Solving linear systems. Examples; how many solutions? =====
+==== More examples ====
 ==== Example 1 ====
+If $ f(x)=ax^2+bx+c$ and $f(1)=3$, $f(2)=2$ and $f(3)=4$, find $f(x)$.
+  * $f(1)=3\implies a+b+c=3$
+  * $f(2)=2\implies 4a+2b+c=2$
+  * $f(3)=4\implies 9a+3b+c=4$
+  * $\begin{gather*}  a+b+c=3\\4a+2b+c=2\\9a+3b+c=4\end{gather*}$
+  * Solve using RREF.
+==== ====
+\begin{align*}\def\go#1#2#3{\left[\begin{smallmatrix}#1\\#2\\#3\end{smallmatrix}\right]}
+\def\ar#1{\\\xrightarrow{#1}&}
+\go{1&1&1&3}{4&2&1&2}{9&3&1&4}
+\xrightarrow{R2\to R2-4R1\text{ and }R3\to R3-9R1}&
+\go{1&1&1&3}{0&-2&-3&-10}{0&-6&-8&-23}
+\ar{R2\to -\tfrac12 R2}
+\go{1&1&1&3}{0&1&\tfrac32&5}{0&-6&-8&-23}
+\ar{R3\to R3+6R2}
+\go{1&1&1&3}{0&1&\tfrac32&5}{0&0&1&7}
+\end{align*}
+  * So far: in REF!
+==== ====
+\begin{align*}
+\go{1&1&1&3}{0&1&\tfrac32&5}{0&0&1&7}
+\xrightarrow{R1\to R1-R3\text{ and }R2\to R2-\tfrac32R3}&
+\go{1&1&0&-4}{0&1&0&-5.5}{0&0&1&7}
+\ar{R1\to R1-R2}
+\go{1&0&0&1.5}{0&1&0&-5.5}{0&0&1&7}
+\end{align*}
+  * So $a=1.5$, $b=-5.5$ and $c=7$
+  * So $f(x)=1.5x^2-5.5x+7$.
+==== Example 2 ====
 Solve the linear system
@@ Line 38: / Line 73: @@
   * No free variables.
-==== Example 2 ====
+==== Example 3 ====
 Solve the linear system
@@ Line 66: / Line 101: @@
   * To detect this: put in REF and find a row $[0~0~\dots~0~1]$.
-==== Example 3 ====
+==== Example 4 ====
 For which value(s) of $k$ does the following linear system have