A fact about ideals and subrings which intersect trivially
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.22 Solution: By the Second Isomorphism Theorem for rings, we have $$(S+I)/I \cong S/(S \cap I) =…
Are the following two systems of linear equations equivalent (1)
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.2 Solution: Yes the two systems are equivalent. We show this by writing each equation of the first system in…
Verify that the set of complex numbers is a subfield of C
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.1 Solution: Let $F=\{x+y\sqrt{2}\mid x,y\in\mb Q\}$. Then we must show six things: $0$ is in $F$ $1$ is in $F$…
Solution to Mathematics for Machine Learning Exercise 4.12
Solution: By SVD (Singular Value Decomposition), we have $A=U\Sigma V^T$. Let $y=V^T x$, then \[\|y\|_2^2=y^Ty=(V^Tx)^TV^Tx=x^TVV^Tx=x^Tx=\|x\|_2^2.\]Then we have\begin{align*}\|A x\|_2^2=&\ (Ax)^T(Ax)=x^TA^TAx\\ = &\ x^T V \begin{bmatrix}\sigma_1^2 & 0 & 0\\ 0 &…
In a ring, sets of left and right annihilators are one-sided ideals
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.22 Solution: We begin with a definition and some lemmas. Definition: Let $R$ be a ring, $A…
Characterize the two-sided ideals of a matrix ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.21 Solution: First, let $I \subseteq M_n(R)$ be a two-sided ideal. Let $J \subseteq R$ consist of…
The intersection of an ideal and a subring is a subideal, but not necessarily vice versa
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.20 Solution: $I \cap S$ is a subring by Exercise 7.1.4, so it suffices to show absorption.…
The set of ideals of a ring is closed under arbitrary intersections
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.18 Solution: (1) In Exercise 7.1.4, we showed that $I \cap J$ is a subring of $R$.…
The union of a chain of ideals is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.19 Solution: In Exercise 2.1.15, we saw that $S \subseteq R$ is an additive subgroup. To show…
Ring homomorphisms map an identity element to an identity or a zero divisor
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.17 Solution: (1) Suppose $\varphi(1_R) = r$, with $r \neq 1$. First, if $r = 0$, then…