Elementary axiomatic theory of rings. Integral domains, fields, characteristic. Subrings and product of rings. Homomorphisms, ideals and quotient rings. Isomorphism theorems. Fields of fractions. 

这是一份Bath巴斯大学MA20217/MA20219作业代写的成功案

Use Cramer’s rule to solve the system
$$
\begin{aligned}
2 x_{1}-x_{2} &=1 \
4 x_{1}+4 x_{2} &=20 .
\end{aligned}
$$
Solution. The coefficient matrix and right-hand-side vectors are
$$
A=\left[\begin{array}{rr}
2 & -1 \
4 & 4
\end{array}\right] \text { and } \mathbf{b}=\left[\begin{array}{r}
1 \
20
\end{array}\right]
$$
so that
$$
\operatorname{det} A=8-(-4)=12
$$
and therefore
$$
x_{1}=\frac{\left|\begin{array}{rr}
2 & 1 \
4 & 20
\end{array}\right|}{\left|\begin{array}{rr}
2 & -1 \
4 & 4
\end{array}\right|}=\frac{36}{12}=3 \quad \text { and } \quad x_{2}=\frac{\left|\begin{array}{rr}
1 & -1 \
20 & 4
\end{array}\right|}{\left|\begin{array}{rr}
2 & -1 \
4 & 4
\end{array}\right|}=\frac{24}{12}=2
$$

MA20217/MA20219 COURSE NOTES ：

Since this holds for all $x$, we conclude that $(f+g)+h=f+(g+h)$, which is the associative law for addition of vectors.

Next, if 0 denotes the constant function with value 0 , then for any $f \in V$ we have that for all $0 \leq x \leq 1$,
$$
(f+0)(x)=f(x)+0=f(x) .
$$
(We don’t write the zero element of this vector space in boldface because it’s customary not to write functions in bold.) Since this is true for all $x$ we have that $f+0=f$, which establishes the additive identity law. Also, we define $(-f)(x)=-(f(x))$ so that for all $0 \leq x \leq 1$,
$$
(f+(-f))(x)=f(x)-f(x)=0,
$$