User:Jw116104: Difference between revisions
Appearance
Content deleted Content added
(34 intermediate revisions by the same user not shown) | |||
Line 1: | Line 1: | ||
==Math 163b== |
|||
==Linear Algebra Extra Credit== |
|||
For MATH 511, Submitted 11/9/2010 |
|||
'''With examples, explain...''' |
|||
===#1=== |
===#1=== |
||
Find <math>I = \int_1^2 x\ln{x}\, \mathrm dx\!</math>. |
|||
'''...how to find matrix inverses using row operations.''' |
|||
Let <math>u = \ln{x}</math> and <math>dv = x\,\mathrm dx\!</math>. |
|||
We augment ''A'' with ''I'', then do row operations until we have ''I''|''A<sup>-1</sup>''. |
|||
Then, <math>du = \frac{1}{x}\, \mathrm dx\!</math> and <math>v = \frac{x^2}{2}</math>. |
|||
<math> |
|||
A = |
|||
\left[\begin{array}{ccc} |
|||
0 & 2 & 4 \\ |
|||
2 & 4 & 2 \\ |
|||
3 & 3 & 1 |
|||
\end{array}\right] |
|||
</math> |
|||
So, |
|||
<math> |
|||
(A|I) = |
|||
\left[\begin{array}{ccc|ccc} |
|||
0 & 2 & 4 & 1 & 0 & 0 \\ |
|||
2 & 4 & 2 & 0 & 1 & 0 \\ |
|||
3 & 3 & 1 & 0 & 0 & 1 |
|||
\end{array}\right]</math> |
|||
<math>I=uv\right|_1^2-\int_1^2 v\,\mathrm du.\!</math> |
|||
<math>\rightarrow |
|||
\left[\begin{array}{ccc|ccc} |
|||
2 & 4 & 2 & 0 & 1 & 0 \\ |
|||
0 & 2 & 4 & 1 & 0 & 0 \\ |
|||
3 & 3 & 1 & 0 & 0 & 1 |
|||
\end{array}\right] |
|||
</math> |
|||
<math>\rightarrow |
|||
\left[\begin{array}{ccc|ccc} |
|||
1 & 2 & 1 & 0 & 1/2 & 0 \\ |
|||
0 & 2 & 4 & 1 & 0 & 0 \\ |
|||
3 & 3 & 1 & 0 & 0 & 1 |
|||
\end{array}\right] |
|||
</math> |
|||
<math>\rightarrow |
|||
\left[\begin{array}{ccc|ccc} |
|||
1 & 2 & 1 & 0 & 1/2 & 0 \\ |
|||
0 & 2 & 4 & 1 & 0 & 0 \\ |
|||
0 & -3 & -2 & 0 & -3/2 & 1 |
|||
\end{array}\right] |
|||
</math> |
|||
<math>\rightarrow |
|||
\left[\begin{array}{ccc|ccc} |
|||
1 & 0 & -3 & -1 & 1/2 & 0 \\ |
|||
0 & 1 & 2 & 1/2 & 0 & 0 \\ |
|||
0 & 0 & 4 & 3/2 & -3/2 & 1 |
|||
\end{array}\right] |
|||
</math> |
|||
<math>\rightarrow |
|||
\left[\begin{array}{ccc|ccc} |
|||
1 & 0 & -3 & -1 & 1/2 & 0 \\ |
|||
0 & 1 & 2 & 1/2 & 0 & 0 \\ |
|||
0 & 0 & 1 & 3/8 & -3/8 & 1/4 |
|||
\end{array}\right] |
|||
</math> |
|||
<math>\rightarrow |
|||
\left[\begin{array}{ccc|ccc} |
|||
1 & 0 & 0 & 1/8 & -5/8 & 3/4 \\ |
|||
0 & 1 & 0 & -1/4 & 3/4 & -1/2 \\ |
|||
0 & 0 & 1 & 3/8 & -3/8 & 1/4 |
|||
\end{array}\right] |
|||
</math> |
|||
<math> |
|||
A^{-1} = |
|||
\left[\begin{array}{ccc} |
|||
1/8 & -5/8 & 3/4 \\ |
|||
-1/4 & 3/4 & -1/2 \\ |
|||
3/8 & -3/8 & 1/4 |
|||
\end{array}\right] |
|||
</math> |
|||
===#2=== |
===#2=== |
||
'''...how to recognize when a square matrix is not invertible using the method from #1.'''<br><br> |
|||
<math> |
|||
A = |
|||
\left[\begin{array}{ccc} |
|||
1 & 2 & 1 \\ |
|||
2 & 1 & -1 \\ |
|||
1 & 5 & 4 |
|||
\end{array}\right] |
|||
</math> |
|||
<math> |
|||
(A|I) = |
|||
\left[\begin{array}{ccc|ccc} |
|||
1 & 2 & 1 & 1 & 0 & 0 \\ |
|||
2 & 1 & -1 & 0 & 1 & 0 \\ |
|||
1 & 5 & 4 & 0 & 0 & 1 |
|||
\end{array}\right]</math> |
|||
<math>\rightarrow |
|||
\left[\begin{array}{ccc|ccc} |
|||
1 & 2 & 1 & 1 & 0 & 0 \\ |
|||
0 & -3 & -3 & -2 & 1 & 0 \\ |
|||
1 & 3 & 3 & -1 & 0 & 1 |
|||
\end{array}\right] |
|||
</math> |
|||
<math>\rightarrow |
|||
\left[\begin{array}{ccc|ccc} |
|||
1 & 2 & 1 & 0 & 1/2 & 0 \\ |
|||
0 & -3 & -3 & 1 & 0 & 0 \\ |
|||
0 & 0 & 0 & -3 & 1 & 1 |
|||
\end{array}\right] |
|||
</math> |
|||
Since the A side of the augmentation now has a zero row, A is not invertible. |
|||
===#3=== |
|||
'''...how to express an invertible matrix as a product of elementary matrices using the method from #1.''' |
|||
# List the row ops used |
|||
# Replace each with its “undo” row operation |
|||
# Convert these to elementary matrices (apply to I) and list left to right |
|||
<math> |
|||
A = |
|||
\left[\begin{array}{ccc} |
|||
0 & 2 & 4 \\ |
|||
2 & 4 & 2 \\ |
|||
3 & 3 & 1 |
|||
\end{array}\right] |
|||
</math> |
|||
'''Step 1''' |
|||
<math>R1 \leftrightarrow R2</math><br><br> |
|||
<math>1/2 R1 \rightarrow R1</math><br><br> |
|||
<math>R3 - 3R1 \rightarrow R3</math><br><br> |
|||
<math>1/2 R2 \rightarrow R2</math><br><br> |
|||
<math>R1 - 2R2 \rightarrow R1</math><br><br> |
|||
<math>R3 + 3R2 \rightarrow R3</math><br><br> |
|||
<math>1/4 R3 \rightarrow R3</math><br><br> |
|||
<math>R2 - 2R3 \rightarrow R2</math><br><br> |
|||
<math>R1 + 3R3 \rightarrow R1</math><br><br> |
|||
'''Step 2''' |
|||
<math>R1 \leftrightarrow R2</math><br><br> |
|||
<math>2R1 \rightarrow R1</math><br><br> |
|||
<math>R3 + 3R1 \rightarrow R3</math><br><br> |
|||
<math>2R2 \rightarrow R2</math><br><br> |
|||
<math>R1 + 2R2 \rightarrow R1</math><br><br> |
|||
<math>R3 - 3R2 \rightarrow R3</math><br><br> |
|||
<math>4R3 \rightarrow R3</math><br><br> |
|||
<math>R2 + 2R3 \rightarrow R2</math><br><br> |
|||
<math>R1 - 3R3 \rightarrow R1</math><br><br> |
|||
'''Step 3''' |
|||
<math> |
|||
\left[\begin{array}{ccc} |
|||
0 & 1 & 0 \\ |
|||
1 & 0 & 0 \\ |
|||
0 & 0 & 1 |
|||
\end{array}\right]\ast\left[\begin{array}{ccc} |
|||
2 & 0 & 0 \\ |
|||
0 & 1 & 0 \\ |
|||
0 & 0 & 1 |
|||
\end{array}\right]\ast\left[\begin{array}{ccc} |
|||
1 & 0 & 0 \\ |
|||
0 & 1 & 0 \\ |
|||
3 & 0 & 1 |
|||
\end{array}\right]\ast\left[\begin{array}{ccc} |
|||
1 & 0 & 0 \\ |
|||
0 & 2 & 0 \\ |
|||
0 & 0 & 1 |
|||
\end{array}\right]<br>\ast\left[\begin{array}{ccc} |
|||
1 & 2 & 0 \\ |
|||
0 & 1 & 0 \\ |
|||
0 & 0 & 1 \end{array}\right]\ast\left[\begin{array}{ccc} |
|||
1 & 0 & 0 \\ |
|||
0 & 1 & 0 \\ |
|||
0 & -3 & 1 |
|||
\end{array}\right]\ast\left[\begin{array}{ccc} |
|||
1 & 0 & 0 \\ |
|||
0 & 1 & 0 \\ |
|||
0 & 0 & 4 |
|||
\end{array}\right]\ast\left[\begin{array}{ccc} |
|||
1 & 0 & 0 \\ |
|||
0 & 1 & 2 \\ |
|||
0 & 0 & 1 |
|||
\end{array}\right]<br>\ast\left[\begin{array}{ccc} |
|||
1 & 0 & -3 \\ |
|||
0 & 1 & 0 \\ |
|||
0 & 0 & 1 |
|||
\end{array}\right] = |
|||
\left[\begin{array}{ccc} |
|||
1/8 & -5/8 & 3/4 \\ |
|||
-1/4 & 3/4 & -1/2 \\ |
|||
3/8 & -3/8 & 1/4 |
|||
\end{array}\right] = A^{-1} |
|||
</math> |
|||
===#4=== |
|||
'''...how to calculate the determinant of a square matrix A.''' |
|||
First we must know that if we have a 2x2 matrix |
|||
<math> |
|||
A = |
|||
\left[\begin{array}{cc} |
|||
a & b \\ |
|||
c & d |
|||
\end{array}\right], |
|||
</math> |
|||
then det(''A'') is the scalar ''ad'' - ''bc''. |
|||
We must also know that ''Ã''<sub>''ij''</sub> is the matrix obtained by deleting row ''i'' and column ''j'' from ''A''. |
|||
Now, the cofactor of the entry of ''A'' in row ''i'', column ''j'' is the scalar (-1)<sup>''i+j''</sup>det(Ã<sub>''ij''</sub>). |
|||
The determinant of A equals the sum of the products of each entry in row 1 of A multiplied by its cofactor. |
|||
So if we have |
|||
<math> |
|||
A = |
|||
\left[\begin{array}{cccc} |
|||
2 & 0 & 0 & 1\\ |
|||
0 & 1 & 3 & -3 \\ |
|||
-2 & -3 & -5 & 2 \\ |
|||
4 & -4 & 4 & -6 |
|||
\end{array}\right],</math> |
|||
<math>|A| = (-1)^2(2)|\tilde{A}_{11}|+(-1)^3(0)|\tilde{A}_{12}|+(-1)^4(0)|\tilde{A}_{13}|+(-1)^5(1)|\tilde{A}_{14}|.</math> |
|||
Therefore, |
|||
<math> |
Find <math>I = \int (x+\ln{x}+e^x)\,\mathrm dx\!</math>. |
||
1 & 3 & -3 \\ |
|||
-3 & -5 & 2 \\ |
|||
-4 & 4 & -6 |
|||
\end{array}\right| + 0 + 0 + (-1)^5(1)\left|\begin{array}{cccc} |
|||
0 & 1 & 3 \\ |
|||
-2 & -3 & -5 \\ |
|||
4 & -4 & 4 |
|||
\end{array}\right| |
|||
</math> |
|||
We would then continue using cofactor expansion on the 3x3 matrices and our problem will only involve taking determinants of 2x2 matrices. |
|||
<math> |
<math>I = \int x\,\mathrm dx\!+\int\ln{x}\,\mathrm dx\!+\int e^x\,\mathrm dx\!</math> |
||
===#5=== |
|||
'''...that for every square matrix ''A'' there exists a square matrix ''B'' such that ''AB'' = (|''A''|)*(''I'') is a square matrix where the diagonal entries are all |""A""| and non-diagonal entries are zero.''' |
|||
<math>I = (\frac{x^2}{2}+c_1) + \int\ln{x}\,\mathrm dx\! + (e^x +c_3)</math> |
|||
Let B be the square matrix with the same dimension as A whose i,j entry is (-1)^(i+j) det(A_{ji}) (called the transpose of the cofactor matrix, or the classical adjoint of A). |
|||
Let <math>u = \ln{x}</math> and <math>dv = \mathrm dx\!</math>. |
|||
Here is an example with a 2x2 matrix: |
|||
Then, <math>du = \frac{1}{x} \mathrm dx\!</math> and <math>v = x</math>. |
|||
<math> |
|||
A = |
|||
\left[\begin{array}{cc} |
|||
1 & 4 \\ |
|||
3 & 2 |
|||
\end{array}\right], |
|||
</math> |
|||
So, |
|||
|A| = 2 - 12 = -10 |
|||
<math>I = (\frac{x^2}{2}+c_1) + uv-\int v\,\mathrm du\! + (e^x +c_3)</math> |
|||
<math> |
|||
A^t = |
|||
\left[\begin{array}{cc} |
|||
1 & 3 \\ |
|||
4 & 2 |
|||
\end{array}\right], |
|||
</math> |
|||
<math>I = (\frac{x^2}{2}+c_1) + (\ln{x})(x)-\int (x)\,(\frac{1}{x}\mathrm dx\!) + (e^x +c_3)</math> |
|||
and the cofactor matrix of A<sup>t</sup> is |
|||
<math>I = (\frac{x^2}{2}+c_1) + (\ln{x})(x)-\int \mathrm dx\! + (e^x +c_3)</math> |
|||
<math> |
|||
B = |
|||
\left[\begin{array}{cc} |
|||
2 & -4 \\ |
|||
-3 & 1 |
|||
\end{array}\right]. |
|||
</math> |
|||
<math>I = (\frac{x^2}{2}+c_1) + (x\ln{x}-x+c_2) + (e^x +c_3)</math> |
|||
<math>I = \frac{x^2}{2} + x\ln{x}-x + e^x + C</math> |
|||
<math> |
|||
AB = |
|||
\left[\begin{array}{cc} |
|||
1 & 3 \\ |
|||
4 & 2 |
|||
\end{array}\right] |
|||
\left[\begin{array}{cc} |
|||
2 & -4 \\ |
|||
-3 & 1 |
|||
\end{array}\right] = \left[\begin{array}{cc} |
|||
-10 & 0 \\ |
|||
0 & -10 |
|||
\end{array}\right]. |
|||
</math> |
Latest revision as of 15:56, 23 April 2012
Math 163b
[edit]#1
[edit]Find .
Let and .
Then, and .
So,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle I=uv\right|_1^2-\int_1^2 v\,\mathrm du.\!}
#2
[edit]Find .
Let and .
Then, and .
So,