Note : In class, we won't go into much detail into the material from 1.4 (systems of linear equations) since we'll do it again later, more systemeatically. Nonetheless, it's important to work through some of these examples because it will help you apprciate the more systematic approach.

• A subfield of a field $F$ is a subset $F'\subset F$, which is also a field, with addition and multiplication given by those of $F$. Prove that a subset $F'\subset F$ is a subfield if and only if it has all the following properties: $0,1\in F'$, for all $a,b\in F'$, the sum $a+b$ and product $ab$ are again in $F'$, and for non-zero $a\in F'$ both the multiplicative inverse $a^{-1}$ and the additive inverse $-a$ lie in $F'$. (Thanks to Jiayi Zhang for pointing out an error in an earlier version.)
• Prove the `basic properties' of vector spaces mentioned in class:
  • $x+y=x'+y\ \ \Rightarrow\ \ x=x'$,
  • $a x=a x',\ a\neq 0\ \ \Rightarrow\ \ x=x'$
  • $a x= a' x,\ x\neq 0 \ \Rightarrow\ \ a=a'$
  • $0\cdot x=0$ for all $x\in V$,
  • $ax=0\ \ \Rightarrow\ \ a=0 \mbox{ or } x=0$
• Learn about direct sums of subspaces : Section 1.3, Problems 23, 25, 26, 28, 30.
• Learn about the quotient space of a vector space $V$ modulo a subspace $W$: Section 1.3, Problem 31.
• The direct sum $V_1\oplus V_2$ of two vector spaces $V_1,V_2$ over $F$ is the set $V_1\times V_2$ (i.e., pairs $(v_1,v_2)$ with $v_1\in V_1$ and $v_2\in V_2$), with addition and scalar multiplication defined componentwise. Show that $V_1\oplus V_2$ with these operations is a vector space over $F$, with $V_1, V_2$ as subspaces. (This notion of direct sum is consistent with the direct sum of subspaces.)
• If $V$ is a vector space over a field $F$, and $F'$ is a subfield of $F$, show that $V$ can be made into a vector space over $F'$. (For example, every vector space over the complex numbers becomes a vector space over the reals.)
• Let $V=\mathbb{C}^n$, and $W=\mathbb{R}^n$ (viewed as $n$-tuples of complex numbers that happen to be real). Show that if we regard $V$ as a vector space over $\mathbb{C}$, then $W$ is not a subspace of $V$. However, if we regard $V$ as a vector space over $\mathbb{R}$, then $W$ is a subspace of $V$.
• Section 1.4, Problems 2,3,12,13,14,15. (As mentioned above, at this point we won't cover much along the lines of problem 2,3, but we'll do it soon, more systematically. It's good to get some practice with this now .)