91Ó°ÊÓ

Prove that the set of \(m \times n\) upper triangular matrices is a subspace of \(\mathrm{M}_{m \times n}(F)\).

Show that a subset \(W\) of a vector space \(V\) is a subspace of \(V\) if and only if \(\operatorname{span}(\mathrm{W})=\mathrm{W}\).

Let \(S\) be a nonempty set and \(F\) a field. Let \(\mathcal{C}(S, F)\) denote the set of all functions \(f \in \mathcal{F}(S, F)\) such that \(f(s)=0\) for all but a finite number of elements of \(S\). Prove that \(\mathcal{C}(S, F)\) is a subspace of \(\mathcal{F}(S, F)\).

The set of all \(n \times n\) matrices having trace equal to zero is a subspace \(W\) of \(M_{n \times n}(F)\) (see Example 4 of Section 1.3). Find a basis for W. What is the dimension of W?

Is the set of all differentiable real-valued functions defined on \(R\) a subspace of \(C(R)\) ? Justify your answer.

The set of all upper triangular \(n \times n\) matrices is a subspace \(\mathrm{W}\) of \(\mathrm{M}_{n \times n}(F)\) (see Exercise 12 of Section 1.3). Find a basis for W. What is the dimension of W?

Prove that a set \(S\) of vectors is linearly independent if and only if each finite subset of \(S\) is linearly independent.

Let \(\mathrm{V}\) denote the set of all \(m \times n\) matrices with real entries; so \(\mathrm{V}\) is a vector space over \(R\) by Example 2. Let \(F\) be the field of rational numbers. Is \(\mathrm{V}\) a vector space over \(F\) with the usual definitions of matrix addition and scalar multiplication?

Let \(C^{n}(R)\) denote the set of all real-valued functions defined on the real line that have a continuous \(n\)th derivative. Prove that \(C^{n}(R)\) is a subspace of \(\mathcal{F}(R, R)\).

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