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## Chapter 10 Exercise B

1. Solution. Because $T_\mathbb{C}$ has no real eigenvalues, if $\lambda$ is an eigenvalue of $T_\mathbb{C}$, then $\overline{\lambda}$ is also an eigenvalue of $T_\mathbb{C}$ with equal multiplicity. The determinant of $T_\mathbb{C}$,…

## Chapter 9 Exercise B

1. Solution: Choose an orthonormal basis of $\mathbb{R}^3$ that puts the matrix of $S$ in the form given by 9.36. Since $\mathcal{M}(S)$ is a $3$-by-$3$ matrix, one of the diagonal…

## Chapter 8 Exercise B

1. Solution: By 8.21 (a), $V = G(0, N)$. Since $G(0, N) = \operatorname{null} N^{\operatorname{dim} V}$ (see 8.11), it follows that $N^{\operatorname{dim} V} = 0$ and so $N$ is nilpotent.…

## Chapter 3 Exercise B

1. Give an example of a linear map $T$ such that $\dim \mathrm{null} T=3$ and $\dim \mathrm{range} T = 2$. Solution: Assume $V$ is 5-dimensional vector space with a basis…

## Chapter 2 Exercise B

1. Find all vector spaces that have exactly one basis. Solution: The only vector spaces is $\{0\}$. For if there is a nonzero vector $v$ in a basis, then we…

## Chapter 1 Exercise B

1. Solution: By definition, we have$(-v)+(-(-v))=0\quad\text{and}\quad v+(-v)=0.$This implies both $v$ and $-(-v)$ are additive inverses of $-v$, by the uniqueness of additive inverse, it follows that $-(-v)=v$. 2. Solution: If…