From ba588862d44335f9ba32a6106fa007bc646bb51f Mon Sep 17 00:00:00 2001 From: Steven Clontz Date: Wed, 18 Feb 2026 10:23:15 -0600 Subject: [PATCH 1/7] Update EV6 class activities to clarify spanning R^n and spanning subspace --- source/linear-algebra/source/02-EV/06.ptx | 83 ++++++++++++++++------- 1 file changed, 58 insertions(+), 25 deletions(-) diff --git a/source/linear-algebra/source/02-EV/06.ptx b/source/linear-algebra/source/02-EV/06.ptx index 9a9c2d06d..40d1ebfaf 100644 --- a/source/linear-algebra/source/02-EV/06.ptx +++ b/source/linear-algebra/source/02-EV/06.ptx @@ -43,19 +43,20 @@ Class Activities +

-Recall from section that a subspace of a vector space is -the result of spanning a set of vectors from that vector space. +In we saw an example of +two linearly independent vectors spanning a planar subspace of \IR^3.

-Recall also that a linearly dependent set contains redundant vectors. For example, -only two of the three vectors in are needed to span -the planar subspace. +Because these independent vectors fail to span \IR^3, they are not +a basis for \IR^3. However, they still span a subspace of +\IR^3...

- +

Consider the subspace of \IR^4 given by W=\vspan\left\{ @@ -68,27 +69,59 @@ the planar subspace.

-

- Mark the column of \RREF\left[\begin{array}{cccc} - 2&2&2&1\\ - 3&0&-3&5\\ - 0&1&2&-1\\ - 1&-1&-3&0 - \end{array}\right] that shows that W's spanning set - is linearly dependent. -

+ +

+Which feature of \RREF\left[\begin{array}{cccc} +2&2&2&1\\ +3&0&-3&5\\ +0&1&2&-1\\ +1&-1&-3&0 +\end{array}\right]= +\left[\begin{array}{cccc} +1&0&-1&0\\ +0&1& 2&0\\ +0&0& 0&1\\ +0&0& 0&0 +\end{array}\right] that shows that W's spanning set +is linearly dependent? +

    +
  1. The third column.
    2. +
  2. The fourth column.
    3. +
  3. The third row.
    4. +
  4. The fourth row.
    5. +
+

+ + +

A.

+

+The third columns lacks a pivot, introducing a free variable that +prevents uniqueness of linear combinations. +

+ -

-What would be the result of removing the vector that gave us this column? -

    -
  1. The set still spans W, and remains linearly dependent.
    2. -
  2. The set still spans W, but is now also linearly independent.
    3. -
  3. The set no longer spans W, and remains linearly dependent.
    4. -
  4. The set no longer spans W, but is now linearly independent.
    5. -
-

- + +

+If we removed the vector that causes this issue, +what could we say about that set of three vectors? +

    +
  1. The set spans the vector space \IR^4, but remains linearly dependent.
    2. +
  2. The set spans subspace W\subset \IR^4, but remains linearly dependent.
    3. +
  3. The set spans subspace W\subset \IR^4, and is now linearly independent.
    4. +
  4. The set not longer spans the subspace W\subset \IR^4, but is now linearly independent.
    5. +
+

+ + +

C.

+

+Because the removed vector was already a linear combination of the others, +we still span W. Now that all vectors yield pivot columns, the set +is now independent. +

+ + From 65347c42441acf6e4bfdd7b5baf45280109cea6a Mon Sep 17 00:00:00 2001 From: Steven Clontz Date: Wed, 18 Feb 2026 10:31:37 -0600 Subject: [PATCH 2/7] Apply suggestion from @StevenClontz --- source/linear-algebra/source/02-EV/06.ptx | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/source/linear-algebra/source/02-EV/06.ptx b/source/linear-algebra/source/02-EV/06.ptx index 40d1ebfaf..22e033a65 100644 --- a/source/linear-algebra/source/02-EV/06.ptx +++ b/source/linear-algebra/source/02-EV/06.ptx @@ -107,8 +107,8 @@ If we removed the vector that causes this issue, what could we say about that set of three vectors?
  1. The set spans the vector space \IR^4, but remains linearly dependent.
    2. -
  2. The set spans subspace W\subset \IR^4, but remains linearly dependent.
    3. -
  3. The set spans subspace W\subset \IR^4, and is now linearly independent.
    4. +
  4. The set spans the subspace W\subset \IR^4, but remains linearly dependent.
    5. +
  5. The set spans the subspace W\subset \IR^4, and is now linearly independent.
  6. The set not longer spans the subspace W\subset \IR^4, but is now linearly independent.

From cf23cb7b9902799f6e2b2593917f1612e675a330 Mon Sep 17 00:00:00 2001 From: Steven Clontz Date: Wed, 18 Feb 2026 12:34:25 -0600 Subject: [PATCH 3/7] tweaks, rewordings, remove stale comments --- source/linear-algebra/source/02-EV/06.ptx | 57 +---------------------- 1 file changed, 2 insertions(+), 55 deletions(-) diff --git a/source/linear-algebra/source/02-EV/06.ptx b/source/linear-algebra/source/02-EV/06.ptx index 22e033a65..ed088b98e 100644 --- a/source/linear-algebra/source/02-EV/06.ptx +++ b/source/linear-algebra/source/02-EV/06.ptx @@ -109,7 +109,7 @@ what could we say about that set of three vectors?
  • The set spans the vector space \IR^4, but remains linearly dependent.
  • The set spans the subspace W\subset \IR^4, but remains linearly dependent.
  • The set spans the subspace W\subset \IR^4, and is now linearly independent.
    • -
  • The set not longer spans the subspace W\subset \IR^4, but is now linearly independent.
    • +
  • The set no longer spans the subspace W\subset \IR^4, but is now linearly independent.

    • @@ -124,10 +124,6 @@ is now independent.     - -rref([2,2,2,1; 3,0,-3,5; 0,1,2,-1; 1,-1,-3,0]) - -

    @@ -178,55 +174,6 @@ is now independent.

    - - - - - - - -

    @@ -281,7 +228,7 @@ T=\left\{ .

    -Thus the basis for a subspace is not unique in general. +Thus a given basis for a subspace need not be unique.

    From 530e1f5a4808f673a59753178cd730e1eeeb3f82 Mon Sep 17 00:00:00 2001 From: Steven Clontz Date: Wed, 18 Feb 2026 12:37:52 -0600 Subject: [PATCH 4/7] add example of subspace dimension --- source/linear-algebra/source/02-EV/06.ptx | 32 +++++++++++++++++++++++ 1 file changed, 32 insertions(+) diff --git a/source/linear-algebra/source/02-EV/06.ptx b/source/linear-algebra/source/02-EV/06.ptx index ed088b98e..517ba2a07 100644 --- a/source/linear-algebra/source/02-EV/06.ptx +++ b/source/linear-algebra/source/02-EV/06.ptx @@ -258,6 +258,22 @@ Thus a given basis for a subspace need not be unique. are all valid bases for \IR^3, and they all contain three vectors.

    +

    + Similarly, + + \setList{ + \left[\begin{array}{c}1\\2\\3\end{array}\right], + \left[\begin{array}{c}-2\\0\\5\end{array}\right] + } + \text{ and } + \setList{ + \left[\begin{array}{c}-1\\2\\8\end{array}\right], + \left[\begin{array}{c}0\\4\\11\end{array}\right] + } + + are both valid bases for the same planar subspace of \IR^3, + and they both contain two vectors. +

    @@ -288,6 +304,22 @@ Thus a given basis for a subspace need not be unique. contains exactly three vectors.

    +

    + Likewise, the planar subspace with the following two bases + + \setList{ + \left[\begin{array}{c}1\\2\\3\end{array}\right], + \left[\begin{array}{c}-2\\0\\5\end{array}\right] + } + \text{ and } + \setList{ + \left[\begin{array}{c}-1\\2\\8\end{array}\right], + \left[\begin{array}{c}0\\4\\11\end{array}\right] + } + + has dimension 2 because any basis for the subspace + will have exactly two vectors. +

    From 3c12992d4b37b537c59af784e7aac2a10c7f6134 Mon Sep 17 00:00:00 2001 From: Steven Clontz Date: Wed, 18 Feb 2026 12:47:55 -0600 Subject: [PATCH 5/7] Apply suggestion from @StevenClontz --- source/linear-algebra/source/02-EV/06.ptx | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/source/linear-algebra/source/02-EV/06.ptx b/source/linear-algebra/source/02-EV/06.ptx index 517ba2a07..a41f18c52 100644 --- a/source/linear-algebra/source/02-EV/06.ptx +++ b/source/linear-algebra/source/02-EV/06.ptx @@ -107,9 +107,9 @@ If we removed the vector that causes this issue, what could we say about that set of three vectors?
    1. The set spans the vector space \IR^4, but remains linearly dependent.
      2. -
    2. The set spans the subspace W\subset \IR^4, but remains linearly dependent.
      3. -
    3. The set spans the subspace W\subset \IR^4, and is now linearly independent.
      4. -
    4. The set no longer spans the subspace W\subset \IR^4, but is now linearly independent.
      5. +
    5. The set spans the subspace W of \IR^4, but remains linearly dependent.
      6. +
    6. The set spans the subspace W of \IR^4, and is now linearly independent.
      7. +
    7. The set no longer spans the subspace W of \IR^4, but is now linearly independent.

    From a83539a15e9be71b5d92268f3dfa94de31f6ce13 Mon Sep 17 00:00:00 2001 From: Steven Clontz Date: Wed, 18 Feb 2026 12:50:48 -0600 Subject: [PATCH 6/7] combine fact+definition for dimension --- source/linear-algebra/source/02-EV/06.ptx | 49 ++--------------------- 1 file changed, 3 insertions(+), 46 deletions(-) diff --git a/source/linear-algebra/source/02-EV/06.ptx b/source/linear-algebra/source/02-EV/06.ptx index a41f18c52..73d8ea1c3 100644 --- a/source/linear-algebra/source/02-EV/06.ptx +++ b/source/linear-algebra/source/02-EV/06.ptx @@ -232,56 +232,13 @@ Thus a given basis for a subspace need not be unique.

    - - +

    Any non-trivial real vector space has infinitely-many different bases, but all the bases for a given vector space are exactly the same size. -

    -

    - For example, - - \setList{\vec e_1,\vec e_2,\vec e_3} - \text{ and } - \setList{ - \left[\begin{array}{c}1\\0\\0\end{array}\right], - \left[\begin{array}{c}0\\1\\0\end{array}\right], - \left[\begin{array}{c}1\\1\\1\end{array}\right] - } - \text{ and } - \setList{ - \left[\begin{array}{c}1\\0\\-3\end{array}\right], - \left[\begin{array}{c}2\\-2\\1\end{array}\right], - \left[\begin{array}{c}3\\-2\\5\end{array}\right] - } - - are all valid bases for \IR^3, and they all contain three vectors. -

    -

    - Similarly, - - \setList{ - \left[\begin{array}{c}1\\2\\3\end{array}\right], - \left[\begin{array}{c}-2\\0\\5\end{array}\right] - } - \text{ and } - \setList{ - \left[\begin{array}{c}-1\\2\\8\end{array}\right], - \left[\begin{array}{c}0\\4\\11\end{array}\right] - } - - are both valid bases for the same planar subspace of \IR^3, - and they both contain two vectors. -

    -     -     - - - -

    - The dimension of a vector space or subspace is equal to the size - of any basis for the vector space. + So we say the dimension of a vector space or subspace is equal to the + size of any basis for the vector space.

    As you'd expect, \IR^n has dimension n. From 0528c8f0a143a1aec997d182769ed9d9102a94f4 Mon Sep 17 00:00:00 2001 From: Steven Clontz Date: Wed, 18 Feb 2026 15:24:00 -0600 Subject: [PATCH 7/7] Apply suggestion from @jkostiuk Co-authored-by: jkostiuk --- source/linear-algebra/source/02-EV/06.ptx | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/source/linear-algebra/source/02-EV/06.ptx b/source/linear-algebra/source/02-EV/06.ptx index 73d8ea1c3..51e83ee9c 100644 --- a/source/linear-algebra/source/02-EV/06.ptx +++ b/source/linear-algebra/source/02-EV/06.ptx @@ -82,7 +82,7 @@ Which feature of \RREF\left[\begin{array}{cccc} 0&1& 2&0\\ 0&0& 0&1\\ 0&0& 0&0 -\end{array}\right] that shows that W's spanning set +\end{array}\right] shows that W's spanning set is linearly dependent?

    1. The third column.