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-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-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-DM.xml b/src2/section-DM.xml index f7a8f43..11fa707 100644 --- a/src2/section-DM.xml +++ b/src2/section-DM.xml @@ -917,11 +917,13 @@ + Calculate a Specific Elementary Matrix

Construct the elementary matrix that will effect the row operation \rowopadd{-6}{2}{3} on a 4\times 7 matrix.

 + Calculate a <m>3\times 3</m> Determinant

Compute the determinant of the matrix\begin{bmatrix} 2&3&-1\\ @@ -931,6 +933,7 @@ + Calculate a <m>4\times 4</m> Determinant

Compute the determinant of the matrix\begin{bmatrix} 3 & 9 & -2 & 4 & 2 \\ diff --git a/src2/section-EE.xml b/src2/section-EE.xml index 5f9edaf..ec5b423 100644 --- a/src2/section-EE.xml +++ b/src2/section-EE.xml @@ -1199,6 +1199,7 @@ + Calculate Eigenvalues of a <m>2\times 2</m> Matrix

Suppose A is the 2\times 2 matrixA=\begin{bmatrix} -5 & 8\\-4 & 7 @@ -1206,11 +1207,13 @@ + Calculate Corresponding Eigenspaces of Previous Question

For each eigenvalue of A, find the corresponding eigenspace.

 + Evaluate a Polynomial Expression of Previous Matrix

For the polynomial p(x)=3x^2-x+2 and A from above, compute p(A).

 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-ILT.xml b/src2/section-ILT.xml index c9c8a0e..1f0f9f6 100644 --- a/src2/section-ILT.xml +++ b/src2/section-ILT.xml @@ -840,16 +840,19 @@ + Why Not Injective?

Suppose \ltdefn{T}{\complex{8}}{\complex{5}} is a linear transformation. Why is T not injective?

 + Kernel of an Injective Linear Transformation

Describe the kernel of an injective linear transformation.

 + Relate Theorem KPI to Theorem PSPHS

should remind you of . Why do we say this?

 diff --git a/src2/section-IVLT.xml b/src2/section-IVLT.xml index 1ee3d72..7d3f0c4 100644 --- a/src2/section-IVLT.xml +++ b/src2/section-IVLT.xml @@ -992,16 +992,19 @@ + Conditions for Invertibility

What conditions allow us to easily determine if a linear transformation is invertible?

 + Isomorphic Vector Spaces

What does it mean to say two vector spaces are isomorphic? Both technically, and informally?

 + Linear Transformations vs Linear Systems

How do linear transformations relate to systems of linear equations?

 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-LT.xml b/src2/section-LT.xml index 51b8c07..b8417f6 100644 --- a/src2/section-LT.xml +++ b/src2/section-LT.xml @@ -1376,16 +1376,19 @@ + Linear Transformation or Not?

    Is the function below a linear transformation? Why or why not?\ltdefn{T}{\complex{3}}{\complex{2}},\quad\lteval{T}{\colvector{x_1\\x_2\\x_3}}=\colvector{3x_1-x_2+x_3\\8x_2-6}.

         + Calculate Matrix Representation

    Determine the matrix representation of the linear transformation S below.\ltdefn{S}{\complex{2}}{\complex{3}},\quad\lteval{S}{\colvector{x_1\\x_2}}=\colvector{3x_1+5x_2\\8x_1-3x_2\\-4x_1}.

         + Theorem LTLC

    has a fairly simple proof. Yet the result itself is very powerful. Comment on why we might say this.

     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 \\ 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-PDM.xml b/src2/section-PDM.xml index 9901341..fbe21a3 100644 --- a/src2/section-PDM.xml +++ b/src2/section-PDM.xml @@ -735,6 +735,7 @@ + Use a Determinant Property

    Consider the two matrices below, and suppose you already have computed \detname{A}=-120. What is \detname{B}? Why? A&= @@ -756,11 +757,13 @@ + Another Nonsingular Equivalence

    State the theorem that allows us to make yet another extension to our NMEx series of theorems.

         + Determinant and Matrix Multiplication

    What is amazing about the interaction between matrix multiplication and the determinant?

     diff --git a/src2/section-PEE.xml b/src2/section-PEE.xml index 4fe626e..ceb5ddd 100644 --- a/src2/section-PEE.xml +++ b/src2/section-PEE.xml @@ -581,16 +581,19 @@ + Eigenvalues of a Nonsingular Matrix

    How can you identify a nonsingular matrix just by looking at its eigenvalues?

         + Maximum Number of Distinct Eigenvalues

    How many different eigenvalues may a square matrix of size n have?

         + Eigenvalues of a Hermitian Matrix

    What is amazing about the eigenvalues of a Hermitian matrix and why is it amazing?

     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-SD.xml b/src2/section-SD.xml index 9b63ff4..5eb36e8 100644 --- a/src2/section-SD.xml +++ b/src2/section-SD.xml @@ -932,16 +932,19 @@ + Equivalence Relation

    What is an equivalence relation?

         + Characterization of Diagonalizable

    State a condition that is equivalent to a matrix being diagonalizable, but is not the definition.

         + Calculate a Diagonalization

    Find a diagonal matrix similar toA=\begin{bmatrix} -5 & 8\\-4 & 7 diff --git a/src2/section-SLT.xml b/src2/section-SLT.xml index 8bcc6d9..6c172af 100644 --- a/src2/section-SLT.xml +++ b/src2/section-SLT.xml @@ -860,16 +860,19 @@ + Why Not Surjective?

    Suppose \ltdefn{T}{\complex{5}}{\complex{8}} is a linear transformation. Why is T not surjective?

         + Range of a Surjective Linear Transformation

    What is the relationship between a surjective linear transformation and its range?

         + Injective vs Surjective

    There are many similarities and differences between injective and surjective linear transformations. Compare and contrast these two different types of linear transformations. (This means going well beyond just stating their definitions.)

     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 ?