Get Your Answer Fast Within An Hour
First time here? Checkout the FAQs!
x

*Math Image Search only works best with zoomed in and well cropped math screenshots. Check DEMO

0 like 0 dislike
75 views
a) Let \(\vec{x}\) and \(\vec{p}\) be points in \(\mathbb{R}^{n}\). Under what conditions on the scalar \(c\) is the set
\[
\|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0
\]
a sphere \(\left\|\vec{x}-\vec{x}_{0}\right\|=R \geq 0\) ? Compute the center, \(\vec{x}_{0}\), and radius, \(R\), in terms of \(\vec{p}\) and \(c\).
b) Let
\[
\begin{aligned}
Q(\vec{x}) &=\sum a_{i j} x_{i} x_{j}+2 \sum b_{i} x_{i}+c \\
&=\langle\vec{x}, A \vec{x}\rangle+2\langle\vec{b}, \vec{x}\rangle+c
\end{aligned}
\]
be a real quadratic polynomial so \(\vec{x}=\left(x_{1}, \ldots, x_{n}\right), \vec{b}=\left(b_{1}, \ldots, b_{n}\right)\) are real vectors and \(A=\left(a_{i j}\right)\) is a real symmetric \(n \times n\) matrix. Just as in the case \(n=1\) (which you should do first), if \(A\) is invertible find a vector \(\vec{v}\) (depending on \(A\) and \(\vec{b}\) ) so that the change of variables \(\vec{y}==\vec{x}-\vec{v}\) (this is a translation by the vector \(\vec{v}\) ) so that in the new \(\vec{y}\) variables \(Q\) has the simpler form
\[
Q=\langle\vec{y}, A \vec{y}\rangle+\gamma \text { that is, } Q=\sum a_{i j} y_{i} y_{j}+\gamma,
\]
where \(\gamma=c-\left\langle\vec{b}, A^{-1} \vec{b}\right\rangle\).
As an example, apply this to \(Q(\vec{x})=2 x_{1}^{2}+2 x_{1} x_{2}+3 x_{2}-4\).
in Mathematics by Platinum (129,882 points) | 75 views

Related questions

0 like 0 dislike
0 answers
0 like 0 dislike
0 answers

Join the MathsGee Study Questions & Answers Club where you get study and financial support for success from our community. SEARCH - ASK - LEARN


On the MathsGee Study Questions & Answers, you can:


1. Ask questions


2. Answer questions


3. Vote on Questions and Answers


4. Tip your favourite community member(s)


5. Create Live Video Tutorials (Paid/Free)


6. Join Live Video Tutorials (Paid/Free)


7. Earn points for participating



Posting on the MathsGee Study Questions & Answers


1. Remember the human


2. Behave like you would in real life


3. Look for the original source of content


4. Search for duplicates before posting


5. Read the community's rules




Q&A RULES


1. Answers to questions will be posted immediately after moderation


2. Questions will be queued for posting immediately after moderation


3. Depending on how many posts we receive, you could be waiting up to 24 hours for your post to appear. But, please be patient as posts will appear after they pass our moderation.


MathsGee Android Q&A