Question

9. Let $Mat_n(\mathbb{R})$ be topologized by identifying it with $\mathbb{R}^{n^2}$ with the standard topology as usual. (a) Show that $Mat_n(\mathbb{Z})$, the set of $n \times n$ matrices with integer entries is a closed, discrete subspace of $Mat_n(\mathbb{R})$. (b) Let G be a compact subgroup of $GL_n(\mathbb{R})$. Show that there are only finitely many matrices in G with all integer entries. (c) A rational number can always be reduced to the form $\frac{m}{n}$ where m and n are relatively prime integers. Such a rational number is said to have denominator bounded by N if $|n| \le N$. Explain why there are only finitely many matrices in the orthogonal group $O(23)$ with rational entries whose denominator is bounded by $10^{20}$.

          9. Let $Mat_n(\mathbb{R})$ be topologized by identifying it with $\mathbb{R}^{n^2}$ with the standard topology as usual.
(a) Show that $Mat_n(\mathbb{Z})$, the set of $n \times n$ matrices with integer entries is a closed, discrete subspace of $Mat_n(\mathbb{R})$.
(b) Let G be a compact subgroup of $GL_n(\mathbb{R})$. Show that there are only finitely many matrices in G with all integer entries.
(c) A rational number can always be reduced to the form $\frac{m}{n}$ where m and n are relatively prime integers. Such a rational number is said to have denominator bounded by N if $|n| \le N$. Explain why there are only finitely many matrices in the orthogonal group $O(23)$ with rational entries whose denominator is bounded by $10^{20}$.

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9. Let Matn(ℝ) be topologized by identifying it with ℝ^n^2 with the standard topology as usual.
(a) Show that Matn(ℤ), the set of n × n matrices with integer entries is a closed, discrete subspace of Matn(ℝ).
(b) Let G be a compact subgroup of GLn(ℝ). Show that there are only finitely many matrices in G with all integer entries.
(c) A rational number can always be reduced to the form (m)/(n) where m and n are relatively prime integers. Such a rational number is said to have denominator bounded by N if |n| ≤ N. Explain why there are only finitely many matrices in the orthogonal group O(23) with rational entries whose denominator is bounded by 10^20.

Added by Elizabeth M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Supreeta N

12/12/2023

Step 1

To show that Matn(Z) is closed in Matn(R), we need to show that its complement is open. Let A be a matrix in Matn(R) that is not in Matn(Z). This means that at least one entry of A is not an integer. Let ε be the distance between this non-integer entry and the Show more…

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Let Mat_(n)(R) be topologized by identifying it with R^(n^(2)) with the standard topology as usual. (a) Show that Mat_(n)(Z) the set of n imes n matrices with integer entries is a closed, discrete subspace of Mat_(n)(R). (b) Let G be a compact subgroup of GL_(n)(R). Show that there are only finitely many matrices in G with all integer entries. (c) A rational number can always be reduced to the form (m)/(n) where m and n are relatively prime integers. Such a rational number is said to have de- nominator bounded by N if |n|<=N. Explain why there are only finitely many matrices in the orthogonal group O(23) with rational entries whose denominator is bounded by 10^(20). 9. Let Matn(R) be topologized by identifying it with Rn? with the standard topology as usual. (a) Show that Matn(Z) the set of n X n matrices with integer entries is a closed, discrete subspace of Matn(R). (b) Let G be a compact subgroup of GLn(R). Show that there are only finitely many matrices in G with all integer entries n are relatively prime integers. Such a rational number is said to have de- nominator bounded by N if[n< N. Explain why there are only finitely many matrices in the orthogonal group O(23) with rational entries whose denominator is bounded by 1020.

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