Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.6 Solution: Write the two systems as follows: $$ \begin{array}{c} a_{11}x+a_{12}y=0\\ a_{21}x+a_{22}y=0\\ \vdots\\ a_{m1}x+a_{m2}y=0 \end{array} \quad\quad \begin{array}{c} b_{11}x+b_{12}y=0\\ b_{21}x+b_{22}y=0\\ \vdots\\…
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.5 Solution: We must check the nine conditions on pages 1-2: An operation is commutative if the table is symmetric…
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.4 Solution: These systems are not equivalent. Call the two equations in the first system $E_1$ and $E_2$ and the…
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.3 Solution: Yes the two systems are equivalent. We show this by writing each equation of the first system in…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.16 Solution: (1) Evidently, $$\overline{g(x)} = \overline{-2x^3 + 25936x + 3}.$$ (2) Note that $$x^4-16 = (x+2)(x-2)(x^2+4).$$…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.14 Solution: We begin with a lemma. Lemma: Let $R$ be a commutative ring with $1 \neq…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.13 Solution: (1) By Exercise 7.3.24, $\varphi^\ast[P]$ is an ideal of $R$. Now suppose $ab \in \varphi^\ast[P]$.…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.12 Solution: ($\subseteq$) Let $x = \sum_i r_is_i \in IJ$, where $$r_i = \sum_j t_{i,j}a_j$$ and $$s_i…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.11 Solution: Suppose $I \not\subseteq P$. Then there exists $a \in I$ such that $a \notin P$.…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.10 Solution: Let $a,b \in R$ such that $ab = 0$. Since $P$ is a prime ideal…