Exercise 1.3.1

Question

Label the following statements as true or false.
\begin{enumerate}[label=(\alph*)]
\item If $\mathrm{V}$ is a vector space and $\mathrm{W}$ is a subset of $\mathrm{V}$ that is a vector space, then $\mathrm{W}$ is a subspace of $\mathrm{V}$.
\item The empty set is a subspace of every vector space.
\item If $\mathrm{V}$ is a vector space other than the zero vector space, then $\mathrm{V}$ contains a subspace $\mathrm{W}$ such that $\mathrm{W} 
eq \mathrm{V}$.
\item The intersection of any two subsets of $\mathrm{V}$ is a subspace of $\mathrm{V}$.
\item An $n 	imes n$ diagonal matrix can never have more than $n$ nonzero entries.
\item The trace of a square matrix is the product of its diagonal entries.
\item Let $\mathrm{W}$ be the $x y$-plane in $\mathrm{R}^{3}$; that is, $\mathrm{W}=\left\{\left(a_{1}, a_{2}, 0ight): a_{1}, a_{2} \in Right\}$. Then $\mathrm{W}=\mathrm{R}^{2}$.
\end{enumerate}

Jephian C.-H. Lin · Accepted Answer

\begin{enumerate}[label=(\alph*)]
\item No. This should make sure that the field and the operations of $V$ and $W$ are the same. Otherwise for example, $V=\mathbb{R}$ and $W=\mathbb{Q}$ respectly. Then $W$ is a vector space over $\mathbb{Q}$ but not a space over $\mathbb{R}$ and so not a subspace of $V$. 
\item No. We should have that any subspace contains $0$.
\item Yes. We can choose $W=0$.
\item No. Let $V=\mathbb{R}$, $E_0=\{0\}$ and $E_1=\{1\}$. Then we have $E_0 \cap E_1=\emptyset$ is not a subspace.
\item Yes. Only entries on diagonal could be nonzero.
\item No. It's the summation of that.
\item No. But it's called isomorphism. That is, they are the same in view of structure.
\end{enumerate}

Exercise 1.3.1

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Exercise 1.3.1

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