Exercises 2.6 Exercises
2.
Find all subspaces in \(\R^2\) that contain the vector \((2,2)\text{.}\)3.
Show that a set with two vectors from \(\R^3\) can not span \(\R^3\text{.}\)4.
Consider a map \(\pi_1\colon\R^n\to\R^n\) given by \((x_1,x_2,\ldots,x_n)\mapsto (x_1,0,\ldots,0)\text{.}\) Show that the image of \(\pi_1\) is a subspace of \(\R^n\text{.}\)5.
Find all subspaces of the vector space \(\R\) over \(\R\text{.}\)6.
Determine whether \((8,-2,-7)\in\R^3\) is a linear combination of the vectors \((1,1,1), (-2,0,1), (-1,3,5)\in\R^3\text{.}\) Here we consider \(\R^3\) as a vector space over \(\R\text{.}\)7.
Consider a vector space \(M_{2\times 2}(F)\) of \(2\times 2\) matrices over a field \(F\text{.}\) Show that the span of
\begin{equation*}
S=\bigg\{\begin{pmatrix}1\amp 0\\0\amp 0\end{pmatrix},\begin{pmatrix}0\amp 1\\0\amp 0\end{pmatrix},\begin{pmatrix}0\amp 0\\1\amp 0\end{pmatrix},\begin{pmatrix}0\amp 0\\0\amp 1\end{pmatrix}\bigg\}
\end{equation*}
is \(M_{2\times 2}(F)\text{.}\)8.
Let \(V\) be a vector space over a field \(F\text{,}\) and let \(S=\{v_1,v_2,v_3\}\subset V\text{.}\) Suppose that \(w=\alpha v_1+\beta v_2\) for some \(\alpha,\beta\in F\text{.}\) Is \(\langle S\rangle=\langle\{w,v_2,v_3\}\rangle\text{?}\)9.
Show that a union \(W_1\cup W_2\) of subspaces of a vector space need not be a subspace of \(V\text{.}\)10.
Let \(V\) be a nontrivial (i.e., \(V\neq\{0\}\)) vector space over an infinite field \(F\text{.}\) Show that \(V\) is not a union of a finite number of proper subspaces.
Hint.
Suppose that \(V\) is a union of a finite number of proper subspaces \(W_i\text{,}\) \(V=W_1\cup W_2\cup\cdots\cup W_r\text{.}\) We may assume that \(W_1\not\subset W_2\cup W_3\cup\cdots\cup W_r\text{.}\) Choose \(w\in W_1\setminus(W_2\cup W_3\cup\cdots\cup W_r)\) and \(v\in V\setminus W_1\text{.}\) Consider the line through \(v\) and parallel to \(w\text{,}\) i.e., the set \(U=\{\alpha w+v:\alpha\in F\}\text{.}\) Show that \(U\) is an infinite set and that each \(W_i\) contains at most one vector from \(U\text{.}\)
