;
Learning starts with a question. Asking is a signal for knowledge request!
First time here? Checkout the FAQs!
x

*Math Image Search only works best with SINGLE, zoomed in, well cropped images of math. No selfies and diagrams please :)

1 like 0 dislike
361 views
Let \(v_{1} \ldots v_{k}\) be vectors in a linear space with an inner product \(\langle,\),\(rangle . Define the\) Gram determinant by \(G\left(v_{1}, \ldots, v_{k}\right)=\operatorname{det}\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).
a) If the \(v_{1} \ldots v_{k}\) are orthogonal, compute their Gram determinant.
b) Show that the \(v_{1} \ldots v_{k}\) are linearly independent if and only if their Gram determinant is not zero.
c) Better yet, if the \(v_{1} \ldots v_{k}\) are linearly independent, show that the symmetric matrix \(\left(\left\langle v_{i}, v_{j}\right\rangle\right)\) is positive definite. In particular, the inequality \(G\left(v_{1}, v_{2}\right) \geq 0\) is the Schwarz inequality.
d) Conversely, if \(A\) is any \(n \times n\) positive definite matrix, show that there are vectors \(v_{1}, \ldots, v_{n}\) so that \(A=\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).
e) Let \(\mathcal{S}\) denote the subspace spanned by the linearly independent vectors \(w_{1} \ldots w_{k} .\) If \(X\) is any vector, let \(P_{\mathcal{S}} X\) be the orthogonal projection of \(X\) into \(\mathcal{S}\), prove that the distance \(\left\|X-P_{\mathcal{S}} X\right\|\) from \(X\) to \(\mathcal{S}\) is given by the formula
\[
\left\|X-Z_{\mathcal{S}} X\right\|^{2}=\frac{G\left(X, w_{1}, \ldots, w_{k}\right)}{G\left(w_{1}, \ldots, w_{k}\right)} .
\]
in Mathematics by Platinum (101k points) | 361 views

Related questions

1 like 0 dislike
0 answers
1 like 0 dislike
1 answer
asked Aug 9, 2021 in Mathematics by Maths-Genie Bronze Status (8.8k points) | 316 views
0 like 0 dislike
1 answer
0 like 0 dislike
1 answer
asked Jul 5, 2022 in Mathematics by AstraNova Diamond (58.4k points) | 165 views
0 like 0 dislike
0 answers
0 like 0 dislike
0 answers
0 like 0 dislike
0 answers

Join MathsGee Q&A, where you get instant answers to your questions from our AI, AstraNova and verified by human experts. We use a combination of generative AI and human experts to provide you the best solutions to your problems.

On the MathsGee Q&A, you can:

1. Get instant answer to your questions

2. Convert image to latex

3. AI-generated answers and insights

4. Get expert-verified answers

5. Vote on questions and answers

6. Tip your favorite community members

7. Join expert live video sessions (Paid/Free)

8. Earn points by participating

9. Take a course

10. Enjoy our interactive learning resources


Posting on the MathsGee Q&A

1. Remember the human

2. Act like you would in real life

3. Find original source of content

4. Check for duplicates before publishing

5. Read the community guidelines


MathsGee 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 the number of messages we receive, you could wait up to 24 hours for your message to appear. But be patient as posts will appear after passing our moderation.


MathsGee Q&A


Acalytica


Social Proof


Web Analytics


Courses