91Ó°ÊÓ

Let \(\beta\) be an ordered basis for a finite-dimensional vector space \(\mathrm{V}\), and let \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) be linear. Prove that, for any nonnegative integer \(k\), \(\left[\mathrm{T}^{k}\right]_{\beta}=\left([\mathrm{T}]_{\beta}\right)^{k} .\)

Let \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) be a linear transformation from an \(n\)-dimensional vector space \(\vee\) to an \(m\)-dimensional vector space \(W\). Let \(\beta\) and \(\gamma\) be ordered bases for \(V\) and \(W\), respectively. Prove that \(\operatorname{rank}(T)=\operatorname{rank}\left(L_{A}\right)\) and that nullity \((\mathrm{T})=\) nullity \(\left(\mathrm{L}_{A}\right)\), where \(A=[\mathrm{T}]_{\beta}^{\gamma}\). Hint: Apply Exercise 17 to Figure \(2.2\).

Let \(A\) be an \(n \times n\) incidence matrix that corresponds to a dominance relation. Determine the number of nonzero entries of \(A\).

Let \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) be linear, \(b \in \mathrm{W}\), and \(K=\\{x \in \mathrm{V}: \mathrm{T}(x)=b\\}\) be nonempty. Prove that if \(s \in K\), then \(K=\\{s\\}+\mathrm{N}(\mathrm{T})\). (See page 22 for the definition of the sum of subsets.) The following definition is used in Exercises 25-28 and in Exercise 31 . Definition. Let \(\mathrm{V}\) be a vector space and \(\mathrm{W}_{1}\) and \(\mathrm{W}_{2}\) be subspaces of \(\mathrm{V}\) such that \(\mathrm{V}=\mathrm{W}_{1} \oplus \mathrm{W}_{2}\). (Recall the definition of direct sum given on page 22.) The function \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) defined by \(\mathrm{T}(x)=x_{1}\) where \(x=x_{1}+x_{2}\) with \(x_{1} \in \mathrm{W}_{1}\) and \(x_{2} \in \mathrm{W}_{2}\), is called the projection of \(\mathrm{V}\) on \(\mathrm{W}_{1}\) or the projection on \(\mathrm{W}_{1}\) along \(\mathrm{W}_{2}\).

Let \(T: R^{2} \rightarrow R^{2}\). Include figures for each of the following parts. (a) Find a formula for \(\mathrm{T}(a, b)\), where \(\mathrm{T}\) represents the projection on the \(y\)-axis along the \(x\)-axis. (b) Find a formula for \(\mathrm{T}(a, b)\), where \(\mathrm{T}\) represents the projection on the \(y\)-axis along the line \(L=\\{(s, s): s \in R\\}\).

Let \(\mathrm{T}: \mathrm{R}^{3} \rightarrow \mathrm{R}^{3}\). (a) If \(\mathrm{T}(a, b, c)=(a, b, 0)\), show that \(\mathrm{T}\) is the projection on the \(x y\) plane along the \(z\)-axis. (b) Find a formula for \(\mathrm{T}(a, b, c)\), where \(\mathrm{T}\) represents the projection on the \(z\)-axis along the \(x y\)-plane. (c) If \(\mathrm{T}(a, b, c)=(a-c, b, 0)\), show that \(\mathrm{T}\) is the projection on the \(x y\)-plane along the line \(L=\\{(a, 0, a): a \in R\\}\).

Using the notation in the definition above, assume that \(T: V \rightarrow V\) is the projection on \(\mathrm{W}_{1}\) along \(\mathrm{W}_{2}\). (a) Prove that \(\mathrm{T}\) is linear and \(\mathrm{W}_{1}=\\{x \in \mathrm{V}: \mathrm{T}(x)=x\\}\). (b) Prove that \(\mathrm{W}_{1}=\mathrm{R}(\mathrm{T})\) and \(\mathrm{W}_{2}=\mathrm{N}(\mathrm{T})\). (c) Describe \(\mathrm{T}\) if \(\mathrm{W}_{1}=\mathrm{V}\). (d) Describe \(T\) if \(W_{1}\) is the zero subspace.

Suppose that \(W\) is a subspace of a finite-dimensional vector space \(V\). (a) Prove that there exists a subspace \(\mathrm{W}^{\prime}\) and a function \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) such that \(\mathrm{T}\) is a projection on \(\mathrm{W}\) along \(\mathrm{W}^{\prime}\). (b) Give an example of a subspace \(W\) of a vector space \(V\) such that there are two projections on \(W\) along two (distinct) subspaces.

$$ \text { Prove that the subspaces }\\{0\\}, V, R(T) \text {, and } N(T) \text { are all } T \text {-invariant. } $$

$$ \text { Prove that the subspaces }\\{0\\}, V, R(T) \text {, and } N(T) \text { are all } T \text {-invariant. } $$