91Ó°ÊÓ

Suppose that \(\mathrm{T}\) is the projection on \(\mathrm{W}\) along some subspace \(\mathrm{W}^{\prime}\). Prove that \(\mathrm{W}\) is \(\mathrm{T}\)-invariant and that \(\mathrm{T}_{\mathrm{W}}=\mathrm{I}_{\mathrm{W}}\).

Prove the following generalization of Theorem 2.6: Let \(V\) and \(W\) be vector spaces over a common field, and let \(\beta\) be a basis for V. Then for any function \(f: \beta \rightarrow \mathrm{W}\) there exists exactly one linear transformation \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) such that \(\mathrm{T}(x)=f(x)\) for all \(x \in \beta .\) Exercises 36 and 37 require the definition of direct sum given on page 22 .

Let \(\mathrm{V}\) be a finite-dimensional vector space and \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) be linear. (a) Suppose that \(\mathrm{V}=\mathrm{R}(\mathrm{T})+\mathrm{N}(\mathrm{T})\). Prove that \(\mathrm{V}=\mathrm{R}(\mathrm{T}) \oplus \mathrm{N}(\mathrm{T})\). (b) Suppose that \(R(T) \cap N(T)=\\{0\\}\). Prove that \(V=R(T) \oplus N(T)\). Be careful to say in each part where finite-dimensionality is used.

Let \(\mathrm{T}: C \rightarrow C\) be the function defined by \(\mathrm{T}(z)=\bar{z}\). Prove that \(\mathrm{T}\) is additive (as defined in Exercise 38 ) but not linear.

Prove that there is an additive function \(\mathrm{T}: R \rightarrow R\) (as defined in Exercise 38) that is not linear. Hint: Let \(\mathrm{V}\) be the set of real numbers regarded as a vector space over the field of rational numbers. By the corollary to Theorem \(1.13\) (p. 61 ), \(\mathrm{V}\) has a basis \(\beta\). Let \(x\) and \(y\) be two distinct vectors in \(\beta\), and define \(f: \beta \rightarrow \mathrm{V}\) by \(f(x)=y, f(y)=x\), and \(f(z)=z\) otherwise. By Exercise 35 , there exists a linear transformation \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) such that \(\mathrm{T}(u)=f(u)\) for all \(u \in \beta\). Then \(\mathrm{T}\) is additive, but for \(c=y / x, \mathrm{~T}(c x) \neq c \mathrm{~T}(x)\).