숙제1

A. Vector space, subspace, linear dependence, basis, dimension, direct sum, quotient space.

\begin{enumerate}
\item
Let $\mathbb{Q}(\sqrt{2})$ be the set of all real numbers
of the form $\alpha+\beta\sqrt{2}$, where
$\alpha$ and $\beta$ are rational.
\begin{enumerate}
\item
Is $\mathbb{Q}(\sqrt{2})$ a field?
\item
What if $\alpha$ and $\beta$ are required to be integers?
\end{enumerate}

\item Review the problems from Spring semester regarding the examples of a set with operations \(+\) and \(\cdot\) which form vector spaces and which do not.

\item Prove that if \(\mathbb{R}\) is considered as a rational vector space, i.e., a vector space with the rational number field \(\mathbb{Q}\) as the scalar field, then a necessary and sufficient condition that the vectors \(1\) and \(\xi\) in \(\mathbb{R}\) be linearly independent is that the real number \(\xi\) be irrational.

\item Is it true that if \(x,y,z\) are linearly independent vectors, then so also are \(x+y\), \(y+z\) and \(z+x\)?

\item

\begin{enumerate}
\item
Prove or disprove the followings(PODF):
The vectors $(\xi_1,\xi_2)$ and $(\eta_1,\eta_2)$
in $\mathbb{C}^2$ are linearly dependent if and only if
$\xi_1\eta_2=\xi_2\eta_1$.

\item
Find a similar necessary and sufficient condition for the
linear dependence of two vectors in $\mathbb{C}^3$.

\item
Is there a set of three linearly independent vectors in
$\mathbb{C}^2$?
\end{enumerate}

\item Is the set \(\mathbb{R}\) of real numbers a finite-dimensional vector space over the field \(\mathbb{Q}\) of rational numbers? (Knowing somthing about cardinal numbers will help.)

\item If \(M\) and \(N\) are subspaces of a vector space \(V\), and if every vector in \(V\) belongs either to \(M\) or \(N\) (or both), then either \(M=V\) or \(N=V\) (or both).

\item Consider the quotient spaces obtained by reducing the space \(\mathcal{P}\) of polynomials modulo various subspaces. If \(M=\mathcal{P}_n\), is \(\mathcal{P}/M\) finite-dimensional? What if \(M\) is the subspace consisting of all even polynomials? What if \(M\) is the subspace consistiong of all polynomials divisible by \(x_n\) (where \(x_n(t)=t^n\))?

\end{enumerate}

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