Paula Good
2022-03-25
Answered

How can I show these polynomials are not co'?

$x,\text{}x-1$ and $x+1$ in the ring ${\mathbb{Z}}_{6}\left[x\right]$

You can still ask an expert for help

Aarlenlsi1

Answered 2022-03-26
Author has **10** answers

Step 1

The ideal$(x-1,\text{}x+1)$ equals the ideal

$(2,\text{}x-1)$ and this is not the whole ring, otherwise

$1\in (2,\text{}x-1)$ , so $1=2f\left(x\right)+(x-1)g\left(x\right)$ .

Now send x to 1 and get a contradiction.

The ideal

Now send x to 1 and get a contradiction.

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Prove that in any group, an element and its inverse have the same order.

asked 2021-01-31

Let

asked 2022-06-24

struggling on the properties of the idempotent matrix, namely for any $n\times n$ matrix, ${A}^{2}=A.$. The projection matrix defined by $M={I}_{n}-A{\left({A}^{T}A\right)}^{-1}{A}^{T}$ is an idempotent matrix. The question is, for any given $n\times m$ ($n>m$) matrix $B,$, do we have

$\begin{array}{rcl}M& =& {I}_{n}-A{\left({A}^{T}A\right)}^{-1}{A}^{T}\\ & =& {I}_{n}-AB{\left({B}^{T}{A}^{T}AB\right)}^{-1}{B}^{T}{A}^{T},\end{array}$

since $AB$ is basically the linear transformation of matrix $A.$

$\begin{array}{rcl}M& =& {I}_{n}-A{\left({A}^{T}A\right)}^{-1}{A}^{T}\\ & =& {I}_{n}-AB{\left({B}^{T}{A}^{T}AB\right)}^{-1}{B}^{T}{A}^{T},\end{array}$

since $AB$ is basically the linear transformation of matrix $A.$

asked 2022-07-11

I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix.

What is the purpose of such a form? I have taken a usual first-course in linear algebra (did another semester with Axler, but I don't claim to be an expert) and have taken abstract algebra (most familiar with group and ring theory) and have briefly skimmed through linear algebra books covering this material, but I don't quite understand the "big picture" idea, i.e., why is this useful in application? One person once told me it is the "most straightforward and useful algorithm for solving systems of linear equations, once you get beyond 3 variables or so," but maybe I'm missing something, since I usually don't see anything like what this person described to me in the linear algebra books I have. Most textbooks I've seen tend to have a more theoretical focus on this topic.

Also, any suggested texts which have good coverage on this topic would be very helpful.

What is the purpose of such a form? I have taken a usual first-course in linear algebra (did another semester with Axler, but I don't claim to be an expert) and have taken abstract algebra (most familiar with group and ring theory) and have briefly skimmed through linear algebra books covering this material, but I don't quite understand the "big picture" idea, i.e., why is this useful in application? One person once told me it is the "most straightforward and useful algorithm for solving systems of linear equations, once you get beyond 3 variables or so," but maybe I'm missing something, since I usually don't see anything like what this person described to me in the linear algebra books I have. Most textbooks I've seen tend to have a more theoretical focus on this topic.

Also, any suggested texts which have good coverage on this topic would be very helpful.

asked 2021-02-21

Grades in three tests in College Algebra are 87,59 and 73.

The final carries double weight.

The average after final is 75

The points needed on the final to average the points to 75 when the final carries double weight.

The final carries double weight.

The average after final is 75

The points needed on the final to average the points to 75 when the final carries double weight.

asked 2021-05-09

Substitute each point (-3, 5) and (2, -1) into the slope-intercept form of a linear equation to write a system of equations. Then use the system to find the equation of the line containing the two points. Explain your reasoning.

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Is the vector space