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taco

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The Linear Algebra Thread
« on: January 21, 2015, 10:14:51 am »
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The Linear Algebra Thread

Utilize this thread and become beast.

*** 6/9/15 : I forgot to bring notes with me... I'll update this next year ;p ***



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« Last Edit: June 09, 2015, 04:18:39 pm by robo »

taco

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Re: The Linear Algebra Thread
« Reply #1 on: January 24, 2015, 02:36:08 pm »
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Quizzes
Time Limit: 10 minutes


Quiz 1:

Code: [Select]
Let us consider the linear system:

       x2 + 2x3 = 3
 x1 + 2x2 + 3x3 = 0
3x1 + 2x2 +  x3 = 0

1. Give the augmented matrix for the above linear system.

2. Transform the augmented matrix into reduced echelon form by elementary row operations.

3. Which columns are pivot columns?

4. Is the linear system consistent or inconsistent? Why?


Quiz 2:

Code: [Select]
                        [[ 1  3  1 ]                   [ 1
                 A =     [ 2  2  1 ]             b =     0
                         [ 3  1  1 ]]                   -1 ]


1. Consider the matrix equation Ax = 0. Find the general solution to the matrix equation and express it in parametric vector form.

2. Consider the matrix equation Ax = b. Find the general solution to the matrix equation and express it in parametric vector form.
   If the equation is inconsistent, write "Inconsistent" and give the reason.
« Last Edit: February 09, 2015, 12:52:46 pm by robo »

adarqui

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Re: The Linear Algebra Thread
« Reply #2 on: January 25, 2015, 09:27:48 pm »
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Quote
Quiz 1:

Code: [Select]
Let us consider the linear system:

       x2 + 2x3 = 3
 x1 + 2x2 + 3x3 = 0
3x1 + 2x2 +  x3 = 0

1. Give the augmented matrix for the above linear system.

2. Transform the augmented matrix into reduced echelon form by elementary row operations.

3. Which columns are pivot columns?

4. Is the linear system consistent or inconsistent? Why?

nice!

I get an inconsistent system when I reduce the augmented matrix.

First I did it by hand on paper, then I translated it to haskell using linear transformations from the Data.Matrix library.

https://github.com/adarqui/Daimyo/blob/master/Haskell/src/Daimyo/LinearAlgebra/Quiz/Problems.hs

https://hackage.haskell.org/package/matrix-0.3.4.1/docs/Data-Matrix.html


Code: [Select]
*Daimyo.LinearAlgebra.Quiz.Problems Daimyo.LinearAlgebra.Quiz.Problems Data.Maybe> p1
Operation: augmented
( 0.0 1.0 2.0 3.0 )
( 1.0 2.0 3.0 0.0 )
( 3.0 2.0 1.0 0.0 )

Operation: swap R1,R2
( 1.0 2.0 3.0 0.0 )
( 0.0 1.0 2.0 3.0 )
( 3.0 2.0 1.0 0.0 )

Operation: R1= -2*R2+R1
(  1.0  0.0 -1.0 -6.0 )
(  0.0  1.0  2.0  3.0 )
(  3.0  2.0  1.0  0.0 )

Operation: R3= -3*R1+R3
(  1.0  0.0 -1.0 -6.0 )
(  0.0  1.0  2.0  3.0 )
(  0.0  2.0  4.0 18.0 )

Operation: R3= -2*R2+R3
(  1.0  0.0 -1.0 -6.0 )
(  0.0  1.0  2.0  3.0 )
(  0.0  0.0  0.0 12.0 )


Here is how you can do transformations using that library:

Code: [Select]
p1'rref =
    let
        m = p1'augmented'matrix
        m'a = switchRows 1 2 m
        m'b = combineRows 1 (-2) 2 m'a
        m'c = combineRows 3 (-3) 1 m'b
        m'd = combineRows 3 (-2) 2 m'c
    in
        ...

As for pivot columns, are there only 2? Since you count pivot columns after you reduce it, I get 2 pivot columns and then the third row becomes 0 = b, so it doesn't count as a pivot column.

pC!

taco

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Re: The Linear Algebra Thread
« Reply #3 on: January 28, 2015, 05:03:38 pm »
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Problems
Linear Algebra and It's Applications


One:
  • 1.1  : 3, 7, 9, 11, 15
  • 1.2  : 1, 3

Two:
  • 1.2  : 7, 13, 17
  • 1.3  : 1, 5, 11, 13, 15

Three:
  • 1.4  : 7, 11, 13, 15
  • 1.5  : 5, 7, 15, 17
  • 1.6  : 3a, 3b, 7, 13a

Four:
  • 1.7  : 1, 7, 11
  • 1.8  : 3, 17, 19
« Last Edit: February 05, 2015, 04:02:54 pm by robo »

taco

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Re: The Linear Algebra Thread
« Reply #4 on: January 28, 2015, 05:20:01 pm »
0
Quiz Solutions


Quiz 1:
Code: [Select]
1. [[ 0  1  2  3 ]
    [ 1  2  3  0 ]
    [ 3  2  1  0 ]]


2. [[ 0  1  2  3 ]    [[ 1  2  3  0 ]    [[ 1  2  3  0 ]   [[ 1  0 -1 -6 ]   [[ 1  0 -1  0 ]
    [ 1  2  3  0 ]  ~  [ 0  1  2  3 ]  ~  [ 0  1  2  3 ] ~  [ 0  1  2  3 ] ~  [ 0  1  2  0 ]
    [ 3  2  1  0 ]]    [ 3  2  1  0 ]]    [ 0 -4 -8  0 ]    [ 0  0  0  12]    [ 0  0  0  1 ]]


3. 1, 2, & 4.

4. The linear system is inconsistent. In equation 3 we are left with 0 = 1, however 0 cannot equal 1 and thus this system is inconsistent.



Quiz 2:

1.
Code: [Select]
[[ 1   3   1 ]    [[ 1   3   1   0 ]    [[ 1   0   1/4   0 ]
 [ 2   2   1 ]  ~  [ 0   1  1/4  0 ]  ~  [ 0   1   1/4   0 ]
 [ 3   1   1 ]]    [ 0   0   0   0 ]]    [ 0   0    0    0 ]]


 x1     + (1/4)x3 = 0
     x2 + (1/4)x3 = 0
         0  = 0


{x1 = (-1/4)x3
{x2 = (-1/4)x3
{x3 is free

    [[x1]    [[(-1/4)x3]      [[-1/4]
x =  [x2]  =  [(-1/4)x3]  = x3 [-1/4]
     [x3]]    [   x3   ]]      [  1 ]]


 x = (x3)v
   = tv    (where t in R)

2.
Code: [Select]
[[ 1   3   1 ]    [[ 1   3   1   1   ]    [[ 1   0   1/4   -1/2 ]
 [ 2   2   1 ]  ~  [ 0   1  1/4  1/2 ]  ~  [ 0   1   1/4    1/2 ]
 [ 3   1   1 ]]    [ 0   0   0   0   ]]    [ 0   0    0      0 ]]


 x1     + (1/4)x3 = -1/2
     x2 + (1/4)x3 =  1/2
       0  =   0


{x1 = -1/2 - (1/4)x3
{x2 =  1/2 - (1/4)x3
{x3 is free

    [[x1]    [[-1/2 - (1/4)x3]    [[-1/2]     [[-1/4]
x =  [x2]  =  [ 1/2 - (1/4)x3]  =  [ 1/2] + x3 [-1/4]
     [x3]]    [      x3      ]]    [  0 ]      [  1 ]]


x = p + (x3)v
  = p + tv   (where t in R)
« Last Edit: February 09, 2015, 01:10:48 pm by robo »