91Ó°ÊÓ

The set of all skew-symmetric \(n \times n\) matrices is a subspace \(\mathrm{W}\) of \(\mathrm{M}_{n \times n}(F)\) (see Exercise 28 of Section 1.3). Find a basis for W. What is the dimension of W?

Prove that a subset \(W\) of a vector space \(V\) is a subspace of \(V\) if and only if \(0 \in \mathrm{W}\) and \(a x+y \in \mathrm{W}\) whenever \(a \in F\) and \(x, y \in W\).

Let \(W_{1}\) and \(W_{2}\) be subspaces of a vector space \(V\). Prove that \(W_{1} \cup W_{2}\) is a subspace of \(V\) if and only if \(W_{1} \subseteq W_{2}\) or \(W_{2} \subseteq W_{1}\).

Let \(\mathrm{V}\) be a vector space having dimension \(n\), and let \(S\) be a subset of \(\mathrm{V}\) that generates \(\mathrm{V}\). (a) Prove that there is a subset of \(S\) that is a basis for V. (Be careful not to assume that \(S\) is finite.) (b) Prove that \(S\) contains at least \(n\) vectors.

Let \(V\) denote the set of all real-valued functions \(f\) defined on the real line such that \(f(1)=0\). Prove that \(\mathrm{V}\) is a vector space with the operations of addition and scalar multiplication defined in Example \(3 .\)

Prove that if \(\mathrm{W}\) is a subspace of a vector space \(\mathrm{V}\) and \(w_{1}, w_{2}, \ldots, w_{n}\) are in \(\mathrm{W}\), then \(a_{1} w_{1}+a_{2} w_{2}+\cdots+a_{n} w_{n} \in \mathrm{W}\) for any scalars \(a_{1}, a_{2}, \ldots, a_{n} .\) Visit goo.gl/KTg35w for a solution.

Let \(f, g, \in \mathcal{F}(R, R)\) be the functions defined by \(f(t)=e^{r t}\) and \(g(t)=e^{s t}\), where \(r \neq s\). Prove that \(f\) and \(g\) are linearly independent in \(\mathcal{F}(R, R)\).

Let \(V\) denote the vector space of sequences in \(R\), as defined in Example 5 of Section 1.2. Show that the set of convergent sequences \(\left(a_{n}\right)\) (that is, those for which \(\lim _{n \rightarrow \infty} a_{n}\) exists) is a subspace of \(\mathrm{V}\).

Prove that a vector space is infinite-dimensional if and only if it contains an infinite linearly independent subset.

Let \(S_{1}\) and \(S_{2}\) be disjoint linearly independent subsets of \(\mathrm{V}\). Prove that \(S_{1} \cup S_{2}\) is linearly dependent if and only if \(\operatorname{span}\left(S_{1}\right) \cap \operatorname{span}\left(S_{2}\right) \neq\\{0\\}\). Visit goo.gl/Fi8Epr for a solution.