rheisf
2021-12-18
Answered

Can someone show me step-by-step how to diagonalize this matrix? Im

You can still ask an expert for help

asked 2021-06-10

Determine whether the given set S is a subspace of the vector space V.

A. V=${P}_{5}$ , and S is the subset of ${P}_{5}$ consisting of those polynomials satisfying p(1)>p(0).

B.$V={R}_{3}$ , and S is the set of vectors $({x}_{1},{x}_{2},{x}_{3})$ in V satisfying ${x}_{1}-6{x}_{2}+{x}_{3}=5$ .

C.$V={R}^{n}$ , and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.

D. V=${C}^{2}(I)$ , and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.

E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.

F. V=${P}_{n}$ , and S is the subset of ${P}_{n}$ consisting of those polynomials satisfying p(0)=0.

G.$V={M}_{n}(R)$ , and S is the subset of all symmetric matrices

A. V=

B.

C.

D. V=

E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.

F. V=

G.

asked 2020-12-28

Write the vector form of the general solution of the given system of linear equations.

${x}_{1}+2{x}_{2}-{x}_{3}=0$

${x}_{1}+{x}_{2}+{x}_{3}=0$

${x}_{1}+3{x}_{2}-3{x}_{3}=0$

asked 2021-06-18

find an equation of the form

asked 2022-06-07

Given the linear system of equations:

$\{\begin{array}{l}{x}_{1}+{x}_{2}+{x}_{3}=n\\ {x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5}=3n\\ 2{x}_{1}+2{x}_{2}+2{x}_{3}+{x}_{4}+{x}_{5}+3{x}_{6}+3{x}_{7}+3{x}_{8}=10n\end{array}$

how many solutions are in $\mathbb{N}\cup \{0\}$?

The solution must not be using sum notation like $\sum y$.

I know how to find the number of solutions to the regular equations like ${x}_{1}+{x}_{2}+{x}_{3}+\cdots =n$ but I'm not sure how to do this for a system of equations. I thought of substituting some $x$'s:

${x}_{1}+{x}_{2}+{x}_{3}=3n-({x}_{4}+{x}_{5})\phantom{\rule{0ex}{0ex}}{x}_{4}+{x}_{5}=10n-2({x}_{1}+{x}_{2}+{x}_{3})-3({x}_{6}+{x}_{7}+{x}_{8})\phantom{\rule{0ex}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{x}_{1}+{x}_{2}+{x}_{3}=3n-(10n-2({x}_{1}+{x}_{2}+{x}_{3})-3({x}_{6}+{x}_{7}+{x}_{8}))\phantom{\rule{0ex}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{x}_{1}+{x}_{2}+{x}_{3}+3({x}_{6}+{x}_{7}+{x}_{8})=7n\phantom{\rule{1em}{0ex}}\ast $

As far as I understand finding the number of solutions for the system is equivalent to finding the number of solutions to the equation *.

The only next step from here I can think of is using generating functions:

$(1+x+{x}^{2}+\dots {)}^{3}(1+{x}^{3}+{x}^{6}+{x}^{9}+\dots {)}^{3}$

and we need to find the coefficient of ${x}^{7n}$.

From the closed form identities we have:

$\sum _{k=0}^{\mathrm{\infty}}{\textstyle (}\genfrac{}{}{0ex}{}{3-1+k}{k}{\textstyle )}{x}^{k}\cdot \sum _{i=0}^{\mathrm{\infty}}{\textstyle (}\genfrac{}{}{0ex}{}{3-1+i}{i}{\textstyle )}{x}^{3i}$

But I have no idea now how to find the coefficient of 7n from here and certainly not without using some kind of sum notation.

$\{\begin{array}{l}{x}_{1}+{x}_{2}+{x}_{3}=n\\ {x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5}=3n\\ 2{x}_{1}+2{x}_{2}+2{x}_{3}+{x}_{4}+{x}_{5}+3{x}_{6}+3{x}_{7}+3{x}_{8}=10n\end{array}$

how many solutions are in $\mathbb{N}\cup \{0\}$?

The solution must not be using sum notation like $\sum y$.

I know how to find the number of solutions to the regular equations like ${x}_{1}+{x}_{2}+{x}_{3}+\cdots =n$ but I'm not sure how to do this for a system of equations. I thought of substituting some $x$'s:

${x}_{1}+{x}_{2}+{x}_{3}=3n-({x}_{4}+{x}_{5})\phantom{\rule{0ex}{0ex}}{x}_{4}+{x}_{5}=10n-2({x}_{1}+{x}_{2}+{x}_{3})-3({x}_{6}+{x}_{7}+{x}_{8})\phantom{\rule{0ex}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{x}_{1}+{x}_{2}+{x}_{3}=3n-(10n-2({x}_{1}+{x}_{2}+{x}_{3})-3({x}_{6}+{x}_{7}+{x}_{8}))\phantom{\rule{0ex}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{x}_{1}+{x}_{2}+{x}_{3}+3({x}_{6}+{x}_{7}+{x}_{8})=7n\phantom{\rule{1em}{0ex}}\ast $

As far as I understand finding the number of solutions for the system is equivalent to finding the number of solutions to the equation *.

The only next step from here I can think of is using generating functions:

$(1+x+{x}^{2}+\dots {)}^{3}(1+{x}^{3}+{x}^{6}+{x}^{9}+\dots {)}^{3}$

and we need to find the coefficient of ${x}^{7n}$.

From the closed form identities we have:

$\sum _{k=0}^{\mathrm{\infty}}{\textstyle (}\genfrac{}{}{0ex}{}{3-1+k}{k}{\textstyle )}{x}^{k}\cdot \sum _{i=0}^{\mathrm{\infty}}{\textstyle (}\genfrac{}{}{0ex}{}{3-1+i}{i}{\textstyle )}{x}^{3i}$

But I have no idea now how to find the coefficient of 7n from here and certainly not without using some kind of sum notation.

asked 2021-09-15

Write the given system of linear equations as a matrix equation of the form
Ax=b.

x1−2x2+3x3=0

2x1+x2−5x3=4

x1−2x2+3x3=0

2x1+x2−5x3=4

asked 2021-05-18

Write the vector form of the general solution of the given system of linear equations.

asked 2021-05-05

Write the given system of linear equations as a matrix equation of the form Ax=b.