From 50a047ac868ad6ef3429f630ce05958dc979f511 Mon Sep 17 00:00:00 2001 From: jkostiuk Date: Fri, 11 Oct 2024 18:59:15 +0000 Subject: [PATCH 1/9] suggested edits --- source/linear-algebra/source/02-EV/07.ptx | 116 ++++++++++++---------- 1 file changed, 65 insertions(+), 51 deletions(-) diff --git a/source/linear-algebra/source/02-EV/07.ptx b/source/linear-algebra/source/02-EV/07.ptx index 56e6e6ef..9de64b0e 100644 --- a/source/linear-algebra/source/02-EV/07.ptx +++ b/source/linear-algebra/source/02-EV/07.ptx @@ -128,13 +128,68 @@ Rewrite this solution space in the form \setBuilder{ a \left[\begin{array}{c

-Rewrite this solution space in the form \vspan\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\}. +Which of these choices best describes the set of two vectors +\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\} +used in this span? +

    +
  1. +

    + The set is linearly dependent. +

    +
    2. +
  2. +

    + The set is linearly independent. +

    +
    3. +
  3. +

    + The set spans all of \IR^4. +

    +
    4. +
  4. +

    + The set fails to span the solution space. +

    +
    5. +
+

+ + + + + + + +

+Consider the homogeneous system of equations + + +2x_1&\,+\,&4x_2&\,+\,&2x_3 &\,-\,&3 x_4 &\,+\,&31x_5&\,+\,&2x_6&\,-\,&16x_7&=& 0 + + +-1x_1&\,-\,&2x_2&\,+\,&4x_3 &\,-\,&x_4 &\,+\,&2x_5&\,+\,&9x_6&\,+\,&3x_7&=& 0 + + +x_1&\,+\,&2x_2&\,+\,&x_3 &\,+\,& x_4 &\,+\,&3x_5&\,+\,&6x_7&\,+\,&7x_7&=& 0 + + +

+ + +

+Find its solution set (a subspace of \IR^4). +

+ + +

+Rewrite this solution space in the form \setBuilder{ a \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right] + b \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]+c \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]+d \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right] }{a,b,c,d \in \IR}.

Which of these choices best describes the set of two vectors -\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\} +\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right],\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right],\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]\right\} used in this span?

  1. @@ -149,7 +204,7 @@ used in this span?

  2. - The set spans all of \IR^4. + The set spans all of \IR^7.

  3. @@ -163,8 +218,15 @@ used in this span? + + row1=[] + row2=[] + row3=[] + rref([row1;row2,row3]) + +

    @@ -194,54 +256,6 @@ used in this span? - - - - -

    -Consider the homogeneous system of equations - - - 2x_1&\,+\,&4x_2&\,+\,& 2x_3&\,-\,&4x_4 &=& 0 - - --2x_1&\,-\,&4x_2&\,+\,&x_3 &\,+\,& x_4 &=& 0 - - -3x_1&\,+\,&6x_2&\,-\,&x_3 &\,-\,&4 x_4 &=& 0 - - -

    -

    -Find a basis for its solution space. -

    -     - - - - - - - -

    -Consider the homogeneous vector equation - - x_1 \left[\begin{array}{c} 2 \\ -2 \\ 3 \end{array}\right]+ - x_2 \left[\begin{array}{c} 4 \\ -4 \\ 6 \end{array}\right]+ - x_3 \left[\begin{array}{c} 2 \\ 1 \\ -1 \end{array}\right]+ - x_4 \left[\begin{array}{c} -4 \\ 1 \\ -4 \end{array}\right]= - \left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right] - -

    -

    -Find a basis for its solution space. -

    -     -     - - - -

    From 6e81096e50f410884e8c6003121224771875026d Mon Sep 17 00:00:00 2001 From: jkostiuk Date: Fri, 11 Oct 2024 19:03:39 +0000 Subject: [PATCH 2/9] adjusted MC options --- source/linear-algebra/source/02-EV/07.ptx | 14 ++++++++++++-- 1 file changed, 12 insertions(+), 2 deletions(-) diff --git a/source/linear-algebra/source/02-EV/07.ptx b/source/linear-algebra/source/02-EV/07.ptx index 9de64b0e..bf8052f6 100644 --- a/source/linear-algebra/source/02-EV/07.ptx +++ b/source/linear-algebra/source/02-EV/07.ptx @@ -144,7 +144,7 @@ used in this span?

  4. - The set spans all of \IR^4. + The set spans the solution space.

  5. @@ -152,6 +152,11 @@ used in this span? The set fails to span the solution space.

    6. +
  6. +

    + The set is a basis of the solution space. +

    +

@@ -204,7 +209,7 @@ used in this span?

  • - The set spans all of \IR^7. + The set spans the solution space.

  • @@ -212,6 +217,11 @@ used in this span? The set fails to span the solution space.

    • +
  • +

    + The set is a basis for the solution space. +

    +

    • From 85bee088a4dd188eaa1c1e7be30cfa741a9d789c Mon Sep 17 00:00:00 2001 From: jkostiuk Date: Fri, 11 Oct 2024 16:33:41 -0400 Subject: [PATCH 3/9] Update source/linear-algebra/source/02-EV/07.ptx Co-authored-by: Steven Clontz --- source/linear-algebra/source/02-EV/07.ptx | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/source/linear-algebra/source/02-EV/07.ptx b/source/linear-algebra/source/02-EV/07.ptx index bf8052f6..bdff25ad 100644 --- a/source/linear-algebra/source/02-EV/07.ptx +++ b/source/linear-algebra/source/02-EV/07.ptx @@ -130,7 +130,7 @@ Rewrite this solution space in the form \setBuilder{ a \left[\begin{array}{c

    Which of these choices best describes the set of two vectors \left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\} -used in this span? +used in this solution space?

    1. From 00c582368c0a0ecdc371fc03ed495857511ef839 Mon Sep 17 00:00:00 2001 From: jkostiuk Date: Fri, 11 Oct 2024 20:47:47 +0000 Subject: [PATCH 4/9] accepted a suggestion and fixed some typos --- source/linear-algebra/source/02-EV/07.ptx | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/source/linear-algebra/source/02-EV/07.ptx b/source/linear-algebra/source/02-EV/07.ptx index bdff25ad..84e3572f 100644 --- a/source/linear-algebra/source/02-EV/07.ptx +++ b/source/linear-algebra/source/02-EV/07.ptx @@ -183,7 +183,7 @@ x_1&\,+\,&2x_2&\,+\,&x_3 &\,+\,& x_4 &\,+\,&3x_5

      -Find its solution set (a subspace of \IR^4). +Find its solution set (a subspace of \IR^7).

      @@ -195,7 +195,7 @@ Rewrite this solution space in the form \setBuilder{ a \left[\begin{array}{c

      Which of these choices best describes the set of two vectors \left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right],\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right],\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]\right\} -used in this span? +used in this solution space?

      1. @@ -314,7 +314,7 @@ solution space? To create a computer-animated film, an animator first models a scene as a subset of \mathbb R^3. Then to transform this three-dimensional visual data for display on a two-dimensional movie screen or television set, -the computer could apply a linear tranformation that maps visual information +the computer could apply a linear transformation that maps visual information at the point (x,y,z)\in\mathbb R^3 onto the pixel located at (x+y,y-z)\in\mathbb R^2.

        @@ -322,7 +322,7 @@ at the point (x,y,z)\in\mathbb R^3 onto the pixel located at

        -What homoegeneous linear system describes the positions (x,y,z) +What homogeneous linear system describes the positions (x,y,z) within the original scene that would be aligned with the pixel (0,0) on the screen?

        From fc756799745b5f2a22a920e7be24f4eba439152c Mon Sep 17 00:00:00 2001 From: Steven Clontz Date: Mon, 14 Oct 2024 10:26:42 -0500 Subject: [PATCH 5/9] Update source/linear-algebra/source/02-EV/07.ptx Co-authored-by: jkostiuk --- source/linear-algebra/source/02-EV/07.ptx | 5 ----- 1 file changed, 5 deletions(-) diff --git a/source/linear-algebra/source/02-EV/07.ptx b/source/linear-algebra/source/02-EV/07.ptx index 84e3572f..85fac405 100644 --- a/source/linear-algebra/source/02-EV/07.ptx +++ b/source/linear-algebra/source/02-EV/07.ptx @@ -147,11 +147,6 @@ used in this solution space? The set spans the solution space.

        2. -
      2. -

        - The set fails to span the solution space. -

        -

      3. The set is a basis of the solution space. From ac4a55dac97588bafa0965cd9a3814ac32a27311 Mon Sep 17 00:00:00 2001 From: jkostiuk Date: Thu, 14 Nov 2024 15:58:52 +0000 Subject: [PATCH 6/9] implemented suggestions: added answers, textcolor --- .vscode/settings.json | 2 +- source/linear-algebra/source/02-EV/07.ptx | 42 ++++++++++++++++------- 2 files changed, 30 insertions(+), 14 deletions(-) diff --git a/.vscode/settings.json b/.vscode/settings.json index 08f5839f..11c90c16 100644 --- a/.vscode/settings.json +++ b/.vscode/settings.json @@ -4,7 +4,7 @@ "xml.fileAssociations": [ { "pattern": "**/source/**.ptx", - "systemId": "/home/vscode/.ptx/2.6.2/core/schema/pretext.rng" + "systemId": "/home/vscode/.vscode-remote/extensions/oscarlevin.pretext-tools-0.21.2/assets/schema/pretext.rng" } ] } diff --git a/source/linear-algebra/source/02-EV/07.ptx b/source/linear-algebra/source/02-EV/07.ptx index 85fac405..7bd4647f 100644 --- a/source/linear-algebra/source/02-EV/07.ptx +++ b/source/linear-algebra/source/02-EV/07.ptx @@ -156,6 +156,11 @@ used in this solution space?

        + +

        + D. +

        +         @@ -207,11 +212,6 @@ used in this solution space? The set spans the solution space.

        4. -
      4. -

        - The set fails to span the solution space. -

        -

      5. The set is a basis for the solution space. @@ -221,13 +221,18 @@ used in this solution space?

        + +

        + D. +

        +         row1=[] row2=[] row3=[] - rref([row1;row2,row3]) + rref([row1;row2;row3]) @@ -242,18 +247,24 @@ used in this solution space? Thus if \setBuilder{ - a \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\end{array}\right] + - b \left[\begin{array}{c} -1 \\ 0 \\ -4 \\ 1 \end{array}\right] + a \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\\0\\0\\0\end{array}\right] + + b \left[\begin{array}{c} -7 \\ 0 \\ -1 \\ 5\\1\\0\\0 \end{array}\right]+ + c \left[\begin{array}{c} -1 \\ 0 \\ -3 \\ -2\\0\\1\\0 \end{array}\right]+ + d \left[\begin{array}{c} 1 \\ 0 \\ -2 \\ -6\\0\\0\\1 \end{array}\right] }{ - a,b \in \IR - } = \vspan\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\end{array}\right], - \left[\begin{array}{c} -1 \\ 0 \\ -4 \\ 1 \end{array}\right] \right\} + a,b,c,d \in \IR + } = \vspan\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\\0\\0\\0\end{array}\right], + \left[\begin{array}{c} -7 \\ 0 \\ -1 \\ 5\\1\\0\\0 \end{array}\right], + \left[\begin{array}{c} -1 \\ 0 \\ -3 \\ -2\\0\\1\\0 \end{array}\right], + \left[\begin{array}{c} 1 \\ 0 \\ -2 \\ -6\\0\\0\\1 \end{array}\right] \right\} is the solution space for a homogeneous system, then \setList{ - \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\end{array}\right], - \left[\begin{array}{c} -1 \\ 0 \\ -4 \\ 1 \end{array}\right] + \left[\begin{array}{c} -2 \\ \textcolor{blue}{1} \\ 0 \\ 0\\\textcolor{blue}{0}\\\textcolor{blue}{0}\\\textcolor{blue}{0}\end{array}\right], + \left[\begin{array}{c} -7 \\ \textcolor{blue}{0} \\ -1 \\ 5\\\textcolor{blue}{1}\\\textcolor{blue}{0}\\\textcolor{blue}{0} \end{array}\right], + \left[\begin{array}{c} -1 \\ \textcolor{blue}{0} \\ -3 \\ -2\\\textcolor{blue}{0}\\\textcolor{blue}{1}\\\textcolor{blue}{0} \end{array}\right], + \left[\begin{array}{c} 1 \\ \textcolor{blue}{0} \\ -2 \\ -6\\\textcolor{blue}{0}\\\textcolor{blue}{0}\\\textcolor{blue}{1} \end{array}\right] } is a basis for the solution space. @@ -299,6 +310,11 @@ solution space? + +

        + A. +

        +         From 33473a967844c5149367e4b087788f4255846f10 Mon Sep 17 00:00:00 2001 From: Drew Lewis <30658947+siwelwerd@users.noreply.github.com> Date: Thu, 14 Nov 2024 09:31:42 -0800 Subject: [PATCH 7/9] Discard changes to .vscode/settings.json --- .vscode/settings.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.vscode/settings.json b/.vscode/settings.json index 11c90c16..08f5839f 100644 --- a/.vscode/settings.json +++ b/.vscode/settings.json @@ -4,7 +4,7 @@ "xml.fileAssociations": [ { "pattern": "**/source/**.ptx", - "systemId": "/home/vscode/.vscode-remote/extensions/oscarlevin.pretext-tools-0.21.2/assets/schema/pretext.rng" + "systemId": "/home/vscode/.ptx/2.6.2/core/schema/pretext.rng" } ] } From 17e749e1547600cd5f7e39ea3e38e1b7f399368d Mon Sep 17 00:00:00 2001 From: jkostiuk Date: Tue, 17 Dec 2024 21:20:41 +0000 Subject: [PATCH 8/9] reorganize statement/answer --- source/linear-algebra/source/02-EV/07.ptx | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/source/linear-algebra/source/02-EV/07.ptx b/source/linear-algebra/source/02-EV/07.ptx index eb8a9bfc..3ecb7311 100644 --- a/source/linear-algebra/source/02-EV/07.ptx +++ b/source/linear-algebra/source/02-EV/07.ptx @@ -127,6 +127,7 @@ Rewrite this solution space in the form \setBuilder{ a \left[\begin{array}{c

        +

        Which of these choices best describes the set of two vectors \left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\} @@ -154,13 +155,16 @@ used in this solution space?

      +       -

      D.

      + + + From f5023c237d458461df61ac81de83f92140a2f154 Mon Sep 17 00:00:00 2001 From: jkostiuk Date: Tue, 17 Dec 2024 22:19:01 +0000 Subject: [PATCH 9/9] further reorganizing --- source/linear-algebra/source/02-EV/07.ptx | 23 ++++++++++++----------- 1 file changed, 12 insertions(+), 11 deletions(-) diff --git a/source/linear-algebra/source/02-EV/07.ptx b/source/linear-algebra/source/02-EV/07.ptx index 3ecb7311..f7d75f23 100644 --- a/source/linear-algebra/source/02-EV/07.ptx +++ b/source/linear-algebra/source/02-EV/07.ptx @@ -156,12 +156,13 @@ used in this solution space?

    -

    D.

    + +     @@ -196,6 +197,7 @@ Rewrite this solution space in the form \setBuilder{ a \left[\begin{array}{c

    +

    Which of these choices best describes the set of two vectors \left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right],\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right],\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]\right\} @@ -223,20 +225,18 @@ used in this solution space?

    -     - +

    D.

    -     + + + + - row1=[] - row2=[] - row3=[] - rref([row1;row2;row3]) @@ -312,13 +312,14 @@ solution space?

    - - - +

    A.

    + + +