From af00831283532f43da4e00b381d45cf943af21ac Mon Sep 17 00:00:00 2001 From: Drew Lewis Date: Tue, 12 May 2026 15:19:00 +0000 Subject: [PATCH 1/3] Section 2.3 tweaks --- source/linear-algebra/source/02-EV/03.ptx | 18 +++++++++++++++--- 1 file changed, 15 insertions(+), 3 deletions(-) diff --git a/source/linear-algebra/source/02-EV/03.ptx b/source/linear-algebra/source/02-EV/03.ptx index 2002e7d93..fbe67f994 100644 --- a/source/linear-algebra/source/02-EV/03.ptx +++ b/source/linear-algebra/source/02-EV/03.ptx @@ -343,7 +343,7 @@ we'll learn what this means more carefully in a later chapter.

A planar subset of \IR^3 compared with the plane \IR^2. - +

A planar subset of \IR^3.

 @@ -362,6 +362,18 @@ p += plot(vector([3,-1]), color="yellow") p + + + + + + + + + + + +
@@ -1108,7 +1120,7 @@ that is, (kx)+(ky)=(kx)(ky). This is verified by the following calculatio

A square matrix M is symmetricsymmetric matrix if, for each index i,j, the entries m_{ij} = m_{ji}. That is, the matrix is itself when reflected over the diagonal from upper left to lower right. -Prove that the set of n \times n symmetric matrices is a subspace of M_{n \times n}. +Prove that the set of n \times n symmetric matrices is a subspace of M_{n,n}.

@@ -1124,7 +1136,7 @@ Decide if each of the following is a subspace. If so, prove it. If not, provide

Give an example of each of these, or explain why it's not possible that such a thing would exist. -

  • A nonempty subset of M_{2 \times 2} that is not a subspace.
    • +
    • A nonempty subset of M_{2,2} that is not a subspace.
    • A set of two vectors in \mathbb{R}^2 that is not a spanning set.

    From 24f21997a59b689cadbf358e084e7e0949a6ecc7 Mon Sep 17 00:00:00 2001 From: Drew Lewis Date: Tue, 12 May 2026 15:20:04 +0000 Subject: [PATCH 2/3] Source files did not get included in last commit --- .../linear-algebra/source/02-EV/prefigure/EV3-R2.xml | 10 ++++++++++ .../source/02-EV/prefigure/EV3-planar-subset.xml | 11 +++++++++++ 2 files changed, 21 insertions(+) create mode 100644 source/linear-algebra/source/02-EV/prefigure/EV3-R2.xml create mode 100644 source/linear-algebra/source/02-EV/prefigure/EV3-planar-subset.xml diff --git a/source/linear-algebra/source/02-EV/prefigure/EV3-R2.xml b/source/linear-algebra/source/02-EV/prefigure/EV3-R2.xml new file mode 100644 index 000000000..f5855bd52 --- /dev/null +++ b/source/linear-algebra/source/02-EV/prefigure/EV3-R2.xml @@ -0,0 +1,10 @@ + + + + + + + + + \ No newline at end of file diff --git a/source/linear-algebra/source/02-EV/prefigure/EV3-planar-subset.xml b/source/linear-algebra/source/02-EV/prefigure/EV3-planar-subset.xml new file mode 100644 index 000000000..fc4b335c7 --- /dev/null +++ b/source/linear-algebra/source/02-EV/prefigure/EV3-planar-subset.xml @@ -0,0 +1,11 @@ + + + + + + + + + + \ No newline at end of file From 6382644637fc68b25469ed5edcf013ae9415bb64 Mon Sep 17 00:00:00 2001 From: Drew Lewis Date: Tue, 12 May 2026 15:22:38 +0000 Subject: [PATCH 3/3] Section 4 --- source/linear-algebra/source/02-EV/04.ptx | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/source/linear-algebra/source/02-EV/04.ptx b/source/linear-algebra/source/02-EV/04.ptx index f0ae851e4..a28996750 100644 --- a/source/linear-algebra/source/02-EV/04.ptx +++ b/source/linear-algebra/source/02-EV/04.ptx @@ -664,7 +664,7 @@ vectors that can form a linearly independent set? Mathematical Writing Explorations

    - Prove the result of , by showing that, given a set S = \{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\} of vectors, S is linearly independent iff the equation x_1\vec{v}_1 + x_2\vec{v}_2 + \ldots\ + x_n\vec{v}_n = \vec{0} is only true when x_1 = x_2 = \cdots = x_n = 0. + Prove the result of , by showing that, given a set S = \{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\} of vectors, S is linearly independent if and only f the equation x_1\vec{v}_1 + x_2\vec{v}_2 + \ldots\ + x_n\vec{v}_n = \vec{0} is only true when x_1 = x_2 = \cdots = x_n = 0.