From 8d3e976e803d9a63286f5dc93f87f8b5a9dd21bf Mon Sep 17 00:00:00 2001 From: Tanaquil18 Date: Sun, 26 Jun 2022 01:38:02 +0000 Subject: [PATCH 1/2] Add titles to reading questions for Chapter M --- src2/section-CRS.xml | 3 +++ src2/section-FS.xml | 3 +++ src2/section-MINM.xml | 3 +++ src2/section-MISLE.xml | 3 +++ src2/section-MM.xml | 3 +++ src2/section-MO.xml | 3 +++ 6 files changed, 18 insertions(+) diff --git a/src2/section-CRS.xml b/src2/section-CRS.xml index 0ac127a..0cfb169 100644 --- a/src2/section-CRS.xml +++ b/src2/section-CRS.xml @@ -1190,6 +1190,7 @@ + Span of a column space

Write the column space of the matrix below as the span of a set of three vectors and explain your choice of method.\begin{bmatrix} 1 & 3 & 1 & 3\\ @@ -1199,11 +1200,13 @@ + Column space of a nonsingular matrix

Suppose that A is an n\times n nonsingular matrix. What can you say about its column space?

 + Vector in row space?

Is the vector \colvector{0\\5\\2\\3} in the row space of the following matrix? Why or why not?\begin{bmatrix} 1 & 3 & 1 & 3\\ diff --git a/src2/section-FS.xml b/src2/section-FS.xml index a603415..9c7a346 100644 --- a/src2/section-FS.xml +++ b/src2/section-FS.xml @@ -989,6 +989,7 @@ + Calculate element of left null space

Find a nontrivial element of the left null space of A.A= \begin{bmatrix} @@ -999,6 +1000,7 @@ + Calculate parts of extended echelon form

Find the matrices C and L in the extended echelon form of A.A= \begin{bmatrix} @@ -1009,6 +1011,7 @@ + Significance of Theorem FS

Why is a great conclusion to ?

 diff --git a/src2/section-MINM.xml b/src2/section-MINM.xml index 2aa5488..0cbffad 100644 --- a/src2/section-MINM.xml +++ b/src2/section-MINM.xml @@ -591,6 +591,7 @@ + Calculate inverse, solve system

Compute the inverse of the coefficient matrix of the system of equations below and use the inverse to solve the system. 4x_1 + 10x_2 &= 12 @@ -599,6 +600,7 @@ + Inverse doesn't exist

In the reading questions for you were asked to find the inverse of the 3\times 3 matrix below.\begin{bmatrix} 2 & 3 & 1\\ @@ -608,6 +610,7 @@ + Unitary or not

Is the matrix A unitary? Why?A=\begin{bmatrix} \frac{1}{\sqrt{22}}\left(4+2i\right) & \frac{1}{\sqrt{374}}\left(5+3i\right) \\ diff --git a/src2/section-MISLE.xml b/src2/section-MISLE.xml index a06cb18..77616ce 100644 --- a/src2/section-MISLE.xml +++ b/src2/section-MISLE.xml @@ -775,6 +775,7 @@ + Calculate inverse of a <m>2\times 2</m> matrix

Compute the inverse of the matrix below.\begin{bmatrix} -2 & 3\\ @@ -783,6 +784,7 @@ + Calculate inverse of a <m>3\times 3</m> matrix

Compute the inverse of the matrix below.\begin{bmatrix} 2 & 3 & 1\\ @@ -792,6 +794,7 @@ + Title of Theorem SS

Explain why has the title it does. (Do not just state the theorem, explain the choice of the title making reference to the theorem itself.)

 diff --git a/src2/section-MM.xml b/src2/section-MM.xml index 2c6bb8d..56b6d9e 100644 --- a/src2/section-MM.xml +++ b/src2/section-MM.xml @@ -960,6 +960,7 @@ + Matrix-vector product

Form the matrix-vector product of \begin{bmatrix} @@ -973,6 +974,7 @@ + Calculate product of two matrices

Multiply together the two matrices below (in the order given). \begin{bmatrix} @@ -991,6 +993,7 @@ + Matrix-vector product form of a linear system

Rewrite the system of linear equations below as a vector equality and using a matrix-vector product. (This question does not ask for a solution to the system. But it does ask you to express the system of equations in a new form using tools from this section.) 2x_1 + 3x_2 - x_3 &= 0 diff --git a/src2/section-MO.xml b/src2/section-MO.xml index e74e2d0..072868f 100644 --- a/src2/section-MO.xml +++ b/src2/section-MO.xml @@ -922,6 +922,7 @@ + Calculate sum and scalar mult of matrices

Perform the following matrix computation.(6) \begin{bmatrix} @@ -939,11 +940,13 @@ + Vector space of matrices

reminds you of what previous theorem? How strong is the similarity?

 + Calculate transpose

Compute the transpose of the matrix below.\begin{bmatrix} 6 & 8 & 4 \\ From a74dd1215be12ffab38d200a52d3d2848a197eb8 Mon Sep 17 00:00:00 2001 From: Tanaquil18 Date: Sun, 26 Jun 2022 01:53:39 +0000 Subject: [PATCH 2/2] Add titles to reading questions for Chapter VS --- src2/section-B.xml | 3 +++ src2/section-D.xml | 3 +++ src2/section-LISS.xml | 3 +++ src2/section-PD.xml | 3 +++ src2/section-S.xml | 3 +++ src2/section-VS.xml | 3 +++ 6 files changed, 18 insertions(+) diff --git a/src2/section-B.xml b/src2/section-B.xml index b2879d5..3bd9547 100644 --- a/src2/section-B.xml +++ b/src2/section-B.xml @@ -706,6 +706,7 @@ + Columns of a nonsingular matrix

The matrix below is nonsingular. What can you now say about its columns?A=\begin{bmatrix} -3 & 0 & 1\\ @@ -715,11 +716,13 @@ + Linear combination of columns

Write the vector \vect{w}=\colvector{6\\6\\15} as a linear combination of the columns of the matrix A above. How many ways are there to answer this question?

 + Why desire orthonormal?

Why is an orthonormal basis desirable?

 diff --git a/src2/section-D.xml b/src2/section-D.xml index 6fc076e..6f4a6a8 100644 --- a/src2/section-D.xml +++ b/src2/section-D.xml @@ -568,16 +568,19 @@ + Calculate dimension of <m>P_6</m>

What is the dimension of the vector space P_6, the set of all polynomials of degree 6 or less?

 + Relate rank and nullity

How are the rank and nullity of a matrix related?

 + Rank of a nonsingular matrix

Explain why we might say that a nonsingular matrix has full rank.

 diff --git a/src2/section-LISS.xml b/src2/section-LISS.xml index 023ecb5..15cdf18 100644 --- a/src2/section-LISS.xml +++ b/src2/section-LISS.xml @@ -435,6 +435,7 @@ + Linear dependence of set of matrices

Is the set of matrices below linearly independent or linearly dependent in the vector space M_{22}? Why or why not?\set{ \begin{bmatrix} @@ -450,6 +451,7 @@ + Span vs. spans

Explain the difference between the following two uses of the term span:

  1. S is a subset of the vector space V and the span of S is a subspace of V.
    2. @@ -458,6 +460,7 @@ + Linear combination of vectors in linearly independent set

    The setS=\set{ \colvector{6\\2\\1},\, diff --git a/src2/section-PD.xml b/src2/section-PD.xml index bb633da..2fa4f8c 100644 --- a/src2/section-PD.xml +++ b/src2/section-PD.xml @@ -378,16 +378,19 @@ + Title of Theorem G

    Why does have the title it does?

         + Theorem RMRT surprising

    Why is so surprising ?

         + Calculate dimensions of the 4 subspaces

    Row-reduce the matrix A to reduced row-echelon form. Without any further computations, compute the dimensions of the four subspaces, (a) \nsp{A}, (b) \csp{A}, (c) \rsp{A} and (d) \lns{A}.A= \begin{bmatrix} diff --git a/src2/section-S.xml b/src2/section-S.xml index 2500b0c..7fc8296 100644 --- a/src2/section-S.xml +++ b/src2/section-S.xml @@ -707,16 +707,19 @@ + Subspace test

    Summarize the three conditions that allow us to quickly test if a set is a subspace.

         + Apply the subspace test

    Consider the set of vectorsW=\setparts{\colvector{a\\b\\c}}{3a-2b+c=5}. Is the set W a subspace of \complex{3}? Explain your answer.

         + Name five subspaces

    Name five general constructions of sets of column vectors (subsets of \complex{m}) that we now know as subspaces.

     diff --git a/src2/section-VS.xml b/src2/section-VS.xml index bf765e6..7e50687 100644 --- a/src2/section-VS.xml +++ b/src2/section-VS.xml @@ -553,16 +553,19 @@ + Complex vector space

    Comment on how the vector space \complex{m} went from a theorem () to an example ().

         + Calculate linear combination in Crazy Vector Space

    In the crazy vector space, C, () compute the linear combination2(3,\,4)+(-6)(1,\,2).

         + Why prove things about the zero vector?

    Suppose that \alpha is a scalar and \zerovector is the zero vector. Why should we prove anything as obvious as \alpha\zerovector=\zerovector such as we did in ?