Recent questions in Algebra

Linear algebraAnswered question

hEorpaigh3tR 2022-12-04

The coordinates of the origin are ...........

A. (0, 1)

B.(0, 0)

C.(0, -1)

D.(1, 0)

A. (0, 1)

B.(0, 0)

C.(0, -1)

D.(1, 0)

Commutative AlgebraAnswered question

Ghillardi4Pi 2022-12-04

Show that the image of ${\mathbb{P}}^{n}\times {\mathbb{P}}^{m}$ under the Segre embedding $\psi $ is actually irreducible.

Linear algebraAnswered question

e3r2a1cakCh7 2022-12-03

The matrix equation is not solved correctly. Expain the mistake and find the correct solution. Assume that the indicated inverses exist.

$XA=B,X={A}^{-1}B$

$XA=B,X={A}^{-1}B$

Linear algebraAnswered question

umthumaL3e 2022-12-02

With T defined by T(x)=Ax, find a vector x such that T(x)=b

$A=\left(\begin{array}{ccc}1& -3& 2\\ 3& -8& 8\\ 0& 1& 2\\ 1& 0& 8\end{array}\right)\phantom{\rule{2em}{0ex}},\phantom{\rule{2em}{0ex}}b=\left(\begin{array}{c}1\\ 6\\ 3\\ 10\end{array}\right)$

What I have done so far is thatI have merged the two matrices into a single augmented matrix. And row reduced it to get:

$\left(\begin{array}{cccc}1& -3& 2& 1\\ 0& 1& 2& 3\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

So does this just mean that the answer to the question is $\mathbf{x}=\left(\begin{array}{c}1\\ 3\\ 0\\ 0\end{array}\right)$

$A=\left(\begin{array}{ccc}1& -3& 2\\ 3& -8& 8\\ 0& 1& 2\\ 1& 0& 8\end{array}\right)\phantom{\rule{2em}{0ex}},\phantom{\rule{2em}{0ex}}b=\left(\begin{array}{c}1\\ 6\\ 3\\ 10\end{array}\right)$

What I have done so far is thatI have merged the two matrices into a single augmented matrix. And row reduced it to get:

$\left(\begin{array}{cccc}1& -3& 2& 1\\ 0& 1& 2& 3\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

So does this just mean that the answer to the question is $\mathbf{x}=\left(\begin{array}{c}1\\ 3\\ 0\\ 0\end{array}\right)$

Linear algebraAnswered question

Baardegem3Gw 2022-12-01

Find the orthogonal projection of a vector v onto U

Let $u$ be a unit vector in ${R}^{n}$ and let $U$ be the subspace spanned by $u$. Show that the orthogonal projection of a vector $v$ onto $U$ is given by

${\mathrm{proj}}_{U}v=(u{u}^{T})v$

and thus that the matrix of this projection is $u{u}^{T}$. What is the rank of $u{u}^{T}$?

Where ${u}^{T}$ is the transpose of $u$. Any help is appreciate! I have no idea how to begin this other than knowing

${\mathrm{proj}}_{U}v=\frac{(v,u)u}{(u,u)}$

Thanks!

Let $u$ be a unit vector in ${R}^{n}$ and let $U$ be the subspace spanned by $u$. Show that the orthogonal projection of a vector $v$ onto $U$ is given by

${\mathrm{proj}}_{U}v=(u{u}^{T})v$

and thus that the matrix of this projection is $u{u}^{T}$. What is the rank of $u{u}^{T}$?

Where ${u}^{T}$ is the transpose of $u$. Any help is appreciate! I have no idea how to begin this other than knowing

${\mathrm{proj}}_{U}v=\frac{(v,u)u}{(u,u)}$

Thanks!

Linear algebraAnswered question

Jay Park 2022-12-01

For vectors $\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$ and scalar $\lambda $

$1)\overrightarrow{w}\times \overrightarrow{v}=-(\overrightarrow{v}\times \overrightarrow{w})\phantom{\rule{0ex}{0ex}}2)(\lambda \overrightarrow{v})\times \overrightarrow{w}=\lambda (\overrightarrow{v}\times \overrightarrow{w})=\overrightarrow{v}\times (\lambda \overrightarrow{w})\phantom{\rule{0ex}{0ex}}3)\overrightarrow{u}\times (\overrightarrow{v}+\overrightarrow{w})=\overrightarrow{u}\times \overrightarrow{v}+\overrightarrow{u}\times \overrightarrow{w}$

use the properties above which cross product properties

$((\overrightarrow{i}+\overrightarrow{j})\times \overrightarrow{i})\times \overrightarrow{j}$

$1)\overrightarrow{w}\times \overrightarrow{v}=-(\overrightarrow{v}\times \overrightarrow{w})\phantom{\rule{0ex}{0ex}}2)(\lambda \overrightarrow{v})\times \overrightarrow{w}=\lambda (\overrightarrow{v}\times \overrightarrow{w})=\overrightarrow{v}\times (\lambda \overrightarrow{w})\phantom{\rule{0ex}{0ex}}3)\overrightarrow{u}\times (\overrightarrow{v}+\overrightarrow{w})=\overrightarrow{u}\times \overrightarrow{v}+\overrightarrow{u}\times \overrightarrow{w}$

use the properties above which cross product properties

$((\overrightarrow{i}+\overrightarrow{j})\times \overrightarrow{i})\times \overrightarrow{j}$

College algebraOpen question

Natalia Del Vecchio2022-11-30

Given that 1${\mathrm{log}}_{a}\left(3\right)\approx 0.61$ and l${\mathrm{log}}_{a}\left(5\right)\approx 0.9$ , evaluate each of the following. Hint: use the properties of logarithms to rewrite the given logarithm in terms of the the logarithms of 3 and 5.

Linear algebraAnswered question

FofieiyK 2022-11-29

Two vectors are given by $A=4.8\hat{i}+6.6\hat{j},B=5.9\hat{i}+5.1\hat{j}$. Find $A\times B$

$1)-33.3k\phantom{\rule{0ex}{0ex}}2)-24.6k\phantom{\rule{0ex}{0ex}}3)-7.2k\phantom{\rule{0ex}{0ex}}4)-14.5k\phantom{\rule{0ex}{0ex}}5)-18.8k$

$1)-33.3k\phantom{\rule{0ex}{0ex}}2)-24.6k\phantom{\rule{0ex}{0ex}}3)-7.2k\phantom{\rule{0ex}{0ex}}4)-14.5k\phantom{\rule{0ex}{0ex}}5)-18.8k$

Linear algebraAnswered question

quakbIi 2022-11-27

In triangles UVW and XYZ, U and Z are right angles, $UV\cong \overline{YZ}$, and $\overline{VW}\cong \overline{XY}$. Which theorem proves these triangles are congruent? A. HA Congruence Theorem B. LA Congruence Theorem C. LL Congruence Theorem D. None of these

Linear algebraAnswered question

Garrett Mclaughlin 2022-11-27

Solve linear inequality

$\frac{x}{4}-\frac{3}{2}\le \frac{x}{2}+1$

$\frac{x}{4}-\frac{3}{2}\le \frac{x}{2}+1$

Abstract algebraAnswered question

hEorpaigh3tR 2022-11-24

Generators of a free group

If G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?

If G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?

Linear algebraAnswered question

Mollie Wise 2022-11-23

Let B be the basis ${\mathbb{P}}_{3}$ consisting of the Hermite polynomials $1,2t,-2+4{t}^{2}$ and $-12t+8{t}^{3}$; and let $p(t)=1+16{t}^{2}-8{t}^{3}$. Find the coordinate vector of p relative to B.

$[p{]}_{B}=\left[\begin{array}{c}-7\\ -6\\ -4\\ -1\end{array}\right]$

$[p{]}_{B}=\left[\begin{array}{c}-7\\ -6\\ -4\\ -1\end{array}\right]$

College algebraOpen question

Геннадий Шишаков2022-11-17

$C=\frac{5}{9}(F-32)$

The equation above shows how temperature *F*, measured in degrees Fahrenheit, relates to a temperature *C*, measured in degrees Celsius. Based on the equation, which of the following must be true?

1. A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of $\frac{5}{9}$ degree Celsius.

2. A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.

3. A temperature increase of $\frac{5}{9}$ degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.

A) I only

B) II only

C) III only

D) I and II only

AlgebraOpen question

Flame World2022-11-15

Graph the polynomial function. Factor first if the expression is not in factored form.

f(x)=x³+7x²+4x-12

Linear algebraAnswered question

atgnybo4fq 2022-11-12

Determining the Rank of a Matrix

$A=\left(\begin{array}{ccc}1& 2& 4\\ 3& 6& 5\end{array}\right)$

$A=\left(\begin{array}{ccc}1& 2& 4\\ 3& 6& 5\end{array}\right)$

College algebraOpen question

Joe Dirt2022-11-10

One of the two tables below shows data that can best be modeled by a linear function, and the other shows data that can best be modeled by a quadratic function. Identify which table shows the linear data and which table shows the quadratic data, and find a formula for each model.

College algebraOpen question

Joe Dirt2022-11-10

One of the two tables below shows data that can best be modeled by a linear function, and the other shows data that can best be modeled by a quadratic function. Identify which table shows the linear data and which table shows the quadratic data, and find a formula for each model.

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