Exercise B9

Chapter 10 Exercise B
1. Solution. Because $T_\mathbb{C}$ has no real eigenvalues, if $\lambda$ is an eigenvalue of $T_\ma...
Chapter 9 Exercise B
1. Solution: Choose an orthonormal basis of $\mathbb{R}^3$ that puts the matrix of $S$ in the form g...
Chapter 8 Exercise B
1. Solution: By 8.21 (a), $V = G(0, N)$. Since $G(0, N) = \operatorname{null} N^{\operatorname{dim} ...
Chapter 7 Exercise B
1. Solution: It is true. Consider the standard orthonormal basis $e_1,e_2,e_3$ of $\mb R^3$. Define ...
Chapter 6 Exercise B
1. Solution: (a) One can easily check that each of the four vectors has norm $\sin^2 \theta + \cos^2...
Chapter 5 Exercise B
1. Solution: (a) Note that \[ (I-T)(I+T+\cdots+T^{n-1})=I-T^n=I \]and \[ (I+T+\cdots+T^{n-1})(I-T)=I...
Chapter 3 Exercise B
1. Solution: Assume $V$ is 5-dimensional vector space with a basis $e_1$, $\cdots$, $e_5$. Define $T...
Chapter 2 Exercise B
1. Solution: The only vector spaces is $\{0\}$. For if there is a nonzero vector $v$ in a basis, the...
Chapter 1 Exercise B
1. Solution: By definition, we have\[(-v)+(-(-v))=0\quad\text{and}\quad v+(-v)=0.\]This implies both...