# HG changeset patch # User Rob Beezer # Date 1354687655 28800 # Node ID bc60749c9a5c4a051bdb7c7242e534f42d79f77d # Parent 3b066adae2d5da70595e2d82d26f0abce5ce261b Edits to save 6 pages in print diff --git a/src/section-EE.xml b/src/section-EE.xml --- a/src/section-EE.xml +++ b/src/section-EE.xml @@ -358,20 +358,18 @@

This example illustrates the proof of , so will employ the same notation as the proof look there for full explanations. It is not meant to be an example of a reasonable computational approach to finding eigenvalues and eigenvectors. OK, warnings in place, here we go.

-

Let - -A=\begin{bmatrix} +

Consider the matrix $A$, and choose the vector $\vect{x}$, + +A\begin{bmatrix} \end{bmatrix} - -and choose - -\vect{x}=\colvector{3\\0\\3\\-5\\4} - + +\vect{x}\colvector{3\\0\\3\\-5\\4} +

It is important to notice that the choice of $\vect{x}$ could be anything, so long as it is not the zero vector. We have not chosen $\vect{x}$ totally at random, but so as to make our illustration of the theorem as general as possible. You could replicate this example with your own choice and the computations are guaranteed to be reasonable, provided you have a computational tool that will factor a fifth degree polynomial for you.

diff --git a/src/section-OD.xml b/src/section-OD.xml --- a/src/section-OD.xml +++ b/src/section-OD.xml @@ -24,15 +24,12 @@ Upper Triangular Matrix

The $n\times n$ square matrix $A$ is upper triangular if $\matrixentry{A}{ij} =0$ whenever $i>j$.

-
Lower Triangular Matrix

The $n\times n$ square matrix $A$ is lower triangular if $\matrixentry{A}{ij} =0$ whenever - -

- +

Obviously, properties of a lower triangular matrices will have analogues for upper triangular matrices. Rather than stating two very similar theorems, we will say that matrices are triangular of the same type as a convenient shorthand to cover both possibilities and then give a proof for just one type.

diff --git a/src/section-PEE.xml b/src/section-PEE.xml --- a/src/section-PEE.xml +++ b/src/section-PEE.xml @@ -320,8 +320,8 @@ -\end{bmatrix}\\ - +\end{bmatrix} + \frac{1}{2} \begin{bmatrix} diff --git a/src/section-TSS.xml b/src/section-TSS.xml --- a/src/section-TSS.xml +++ b/src/section-TSS.xml @@ -563,7 +563,7 @@ -

For Exercises C21C28, find the solution set of the given system of linear equations. Identify the values of $n$ and $r$, and compare your answers to the results of the theorems of this section.

+

For Exercises C21C28, find the solution set of the system of linear equations. Give the values of $n$ and $r$, and interpret your answers in light of the theorems of this section.

@@ -925,7 +925,7 @@ -If you cannot conclude anything about an entry, then say so. (See for inspiration.) +If you cannot conclude anything about an entry, then say so. (See .) diff --git a/src/section-VR.xml b/src/section-VR.xml --- a/src/section-VR.xml +++ b/src/section-VR.xml @@ -695,14 +695,10 @@ } -and apply $\vectrepname{B}$ to each vector in the linear combination. This gives us a new computation, now in the vector space $\complex{6}$, - -6\colvector{3\\-2\\0\\7\\4\\-3}+2\colvector{-1\\4\\-2\\3\\8\\5} - -which we can compute with operations in $\complex{6}$ (, ), to arrive at - -\colvector{16\\-4\\-4\\48\\40\\-8} - +and apply $\vectrepname{B}$ to each vector in the linear combination. This gives us a new computation, now in the vector space $\complex{6}$, which we can compute with operations in $\complex{6}$ (, ), + +6\colvector{3\\-2\\0\\7\\4\\-3}+2\colvector{-1\\4\\-2\\3\\8\\5}=\colvector{16\\-4\\-4\\48\\40\\-8} +

We are after the result of a computation in $M_{32}$, so we now can apply $\ltinverse{\vectrepname{B}}$ to obtain a $3\times 2$ matrix,