 # Get college algebra help

Recent questions in Algebra Mauricio Hayden 2022-05-24 Answered

### Acos 90 degree matrix transformation.I'm writing a program that transforms a matrix of points by 90°. In it, I have two vectors from which I am performing the rotation. Both vectors are normalized:$A:x:\sqrt{\left(}\frac{1}{3}\right),y:\sqrt{\left(}\frac{1}{3}\right),z:-\sqrt{\left(}\frac{1}{3}\right)\phantom{\rule{0ex}{0ex}}B:x:\sqrt{\left(}\frac{1}{3}\right),y:\sqrt{\left(}\frac{1}{3}\right),z:\sqrt{\left(}\frac{1}{3}\right)$As I visualize it, these two vectors are separated by 90°, but the dot product of these vectors comes out to $\frac{1}{3}$$\sqrt{\left(}\frac{1}{3}\right)\ast \sqrt{\left(}\frac{1}{3}\right)+\sqrt{\left(}\frac{1}{3}\right)\ast \sqrt{\left(}\frac{1}{3}\right)+\sqrt{\left(}\frac{1}{3}\right)\ast -\sqrt{\left(}\frac{1}{3}\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{3}+\frac{1}{3}-\frac{1}{3}\phantom{\rule{0ex}{0ex}}=\frac{1}{3}\phantom{\rule{0ex}{0ex}}$My code is then supposed to use arc-cos to come up with 90° from this number, but I believe arc-cos needs an input of 0 in order to produce a result of 90°. What am I missing here? cricafh 2022-05-24 Answered

### Let $\mathcal{g}$ be a Lie algebra and let $a,b,c\in \mathcal{g}$ be such that $ab=ba$ and $\left[a,b\right]=c\ne 0$. Let . How to prove that $\mathcal{h}$ is isomorphic to the strictly upper triangular algebra $\mathcal{n}\left(3,F\right)$?Problem: If $\mathcal{h}\cong n\left(3,F\right)$ then $\mathrm{\exists }{a}^{\prime },{b}^{\prime },{c}^{\prime }\in \mathcal{n}\left(3,F\right)$ with ${a}^{\prime }{b}^{\prime }={b}^{\prime }{a}^{\prime }$ and $\left[{a}^{\prime },{b}^{\prime }\right]={c}^{\prime }$ as in $h$ But then ${c}^{\prime }$ must equal $0$ whereas $c\in h$ is not $0$? Nylah Burnett 2022-05-24 Answered

### Let $R$ be a commutative finite dimensional $K$-algebra over a field $K$ (for example the monoid ring of a a finite monoid over a field). Assume we have $R$ in GAP. Then we can check whether $R$ is semisimple using the command RadicalOfAlgebra(R). When the value is 0, $R$ is semisimple. Thus $R$ can be written as a finite product of finite field extensions of $K$.Question: Can we obtain those finite field extensions of $K$ or at least their number and $K$-dimensions using GAP? Thomas Hubbard 2022-05-24 Answered

### What is the Z-score for a 10% confidence level (i.e. 0.1 pvalue)?I want the standard answer used for including in my thesis write up. I googled and used excel to calculate as well but they are all slightly different.Thanks. Hailey Newton 2022-05-23 Answered

### If $\mathcal{A}$ is a commutative ${C}^{\ast }$-subalgebra of $\mathcal{B}\left(\mathcal{H}\right)$, where $\mathcal{H}$ is a Hilbert space, then the weak operator closure of $\mathcal{A}$ is also commutative.I can not prove this. Waylon Ruiz 2022-05-23 Answered

### Is there any simple proof (one that does not use continuous functional calculus) for the statement that $\sigma \left({x}^{\ast }x\right)\subseteq \left[0,\mathrm{\infty }\right)$ for any $x\in \mathcal{A}$ where $\mathcal{A}$ is a commutative ${C}^{\ast }$-Algebra? Aiden Barry 2022-05-23 Answered

### My book demonstrates how to test whether a set of vectors $S=\left\{{v}_{1},{v}_{2},{v}_{3}\right\}$ is linearly independent by writing ${c}_{1}{v}_{1}+{c}_{2}{v}_{2}+{c}_{3}{v}_{3}=0$, equating corresponding components to form a system of linear equations, and then reducing the augmented matrix of this system using Gauss-Jordan elimination.My question is, couldn't you just evaluate the determinant of the matrix rather than using elimination? Couldn't you say that if the determinant is nonzero, the system has only the trivial solution and therefore S is independent; and if it's zero, the system has infinitely many solutions (since we know it can't have no solutions as it has at least the trivial solution) and S is therefore dependent? Or do you have to use Gaussian elimination? groupweird40 2022-05-23 Answered

### At school, or in a first-year course on DEs, we learn (perhaps in less abstract language) that if you have a linear $n$th-order differential equation$Ly=f$then the general solution is something of the form$y={a}_{1}{y}_{1}+...+{a}_{n}{y}_{n}+g$where the ${y}_{i}$ are independent and satisfy $L{y}_{i}=0$, and $g$ satisfies $Lg=f$. Then we receive lots of training in how to find the ${y}_{i}$ and $g$.Obviously any choice of the ${a}_{i}$ will give us a solution to $Ly=f$. But how do you know that there aren't any more solutions?We justify this by making an analogy with systems of linear equations $Ax=b$, saying something along the lines of 'the space of solutions has the same dimension as the kernel of $A$'. But that works in finite dimensions - how do we know that the same is true with linear operators? shelohz0 2022-05-23 Answered

### Let $ß=\left({b}_{1},...,{b}_{n}\right)$ be a basis of the vector space $V$, let $T:V\to V$ be a linear transformation of $V$, and let B be the $ß$-matrix of $T$.(a) Prove that $v\in \mathrm{ker}\left(T\right)$ if and only if $\left[v{\right]}_{B}\in \mathrm{ker}\left(B\right)$.(b) Prove that $v\in \text{Im(T)}$ if and only if $\left[v{\right]}_{B}\in \text{Im(T)}$.(c) Prove that $T$ is an isomorphism if and only if $B$ is invertible. ownerweneuf 2022-05-23 Answered

### Given a matrix transformation $\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, why does it go from ${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{2}$, when it squishes the plane into a line? Aidyn Cox 2022-05-23 Answered

### Find a matrix for the linear transformation $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ that first rotates a vector by an angle of 30 deegres in the counterclockwise direction about the y-axis, then reflects the vector about the xz-plane, and lastly projects the vector onto the yz-plane.How would you illustrate a matrix transformation reflecting about the xz-plane in an ${\mathbb{R}}^{3}$ space? Emilio Guzman27 2022-05-22

###  Davin Fields 2022-05-22 Answered

### Linear transformation matrix derivation$A=\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]\in {\mathbb{R}}^{2×2}$Find the transformation matrix with respect to the basis Jaidyn Bush 2022-05-22 Answered

### I have a question regarding matrix transformations. I am trying to transform a parallelogram into the unit square. I need to find a series of matrices which transforms each point.For example, how do I find a, b, c and d (matrix) when (0,1) is part of the unit square and (3,10) is the original point?[a b] [3 ] =  [c d] rs450nigglix2 2022-05-22 Answered

### Househbolder transformation identity matrix dimensionsWhen performing a householder transformation and generating an elementary reflector matrix of the form:$H=I-2\frac{v{v}^{T}}{{v}^{T}v}$How do we know the dimensions of the identity matrix? Jaidyn Bush 2022-05-22 Answered

### $T\left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\right)=3a-4b+c-d$For a transformation such as this, how does one determine the form of the kernel? Is it simply making the right side equal to zero, solving for each individual variable, and then creating a matrix with these 'new' a,b,c,d values? Kaeden Woodard 2022-05-22 Answered

### Qualitatively (or mathematically "light"), could someone describe the difference between a matrix and a tensor? I have only seen them used in the context of an undergraduate, upper level classical mechanics course, and within that context, I never understood the need to distinguish between matrices and tensors. They seemed like identical mathematical entities to me.Just as an aside, my math background is roughly the one of a typical undergraduate physics major (minus the linear algebra). Kendrick Pierce 2022-05-22 Answered

### my problem came about when applying a 2x2 matrix to a hyperbola..I'm fine with transforming point what I have trouble with is transforming lines.Theres two types I'd like help with:(A) : Given a line set to a constant...like y = a, how to I apply this to a matrix? I was thinking of making a column vector with the top part set to zero and the bottom set to a and applying this to the transformation matrix.(B) : For that hyperbola example, it was a simple unit hyperbola with its asymptotes : y=x, y = -x.How do I intepret what these transformations do to lines, with respect to arbitrary constants like y=a and for variables. wanaopatays 2022-05-22 Answered

### In general, we have functors ${\mathcal{S}\mathcal{C}\mathcal{R}}_{R/}\stackrel{\varphi }{\to }{\mathcal{D}\mathcal{G}\mathcal{A}}_{R}\stackrel{\psi }{\to }{\mathcal{E}\mathcal{I}}_{R/}$. If $R$ is a $\mathbf{Q}$-algebra, then $\psi$ is an equivalence of $\mathrm{\infty }$-categories, $\varphi$ is fully faithful, and the essential image of $\varphi$ consists of the connective objects of ${\mathcal{D}\mathcal{G}\mathcal{A}}_{R}\simeq {\mathcal{E}\mathcal{I}}_{R/}$ (that is, those algebras $A$ having ${\pi }_{i}A=0$ for $i<0$).What is the explicit functor $\varphi :{\mathcal{S}\mathcal{C}\mathcal{R}}_{R/}\to {\mathcal{D}\mathcal{G}\mathcal{A}}_{R}$? I suppose that the natural thing would be to take a simplicial $R$-algebra $A$ and assign it to$\begin{array}{r}\varphi \left(A\right)=\underset{i=0}{\overset{\mathrm{\infty }}{⨁}}{\pi }_{i}A,\end{array}$and take a map $f:A\to B$ and assign it to$\begin{array}{r}\varphi \left(f\right)=\underset{i=0}{\overset{\mathrm{\infty }}{⨁}}\left({f}_{i}:{\pi }_{i}A\to {\pi }_{i}B\right),\end{array}$but as far as I could find this isn't stated explicitly in DAG. Is this the case, and if so, do you have a source or proof? And how does one show that $⨁{\pi }_{i}A$ is a differential graded algebra? infogus88 2022-05-22 Answered

### Question:Find the $3×3$ matrix A, associated with the linear transformation that projects vectors in ${\mathbb{R}}^{\mathbb{3}}$ (orthogonally) onto the plane $x+y+z=0$.The matrix:$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

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