Exercise 9.2.1 Solution: (a) No. Since $f(0,\beta)\ne 0$. (b) No. Since $f((0,0),(1,0))\ne 0$. (c) Yes. Since $f(\alpha,\beta)=4x_1\bar y_1$. (d) No. Because of $\bar x_2$ there, it is not linear on…
Exercise 7.5.1 If $N$ is a nilpotent linear operator on $V$, show that for any polynomial $f$ the semi-simple part of $f(N)$ is a scalar multiple of the identity operator…
Exercise 7.1.1 Let $T$ be a linear operator on $F^2$. Prove that any non-zero vector which is not a characteristic vector for $T$ is a cyclic vector for $T$. Hence,…
Exercise 6.8.3 If $V$ is the space of all polynomials of degree less than or equal to $n$ over a field $F$, prove that the differentiation operator on $V$ is…
Exercise 6.6.1 Let $V$ be a finite-dimensional vector space and let $W_1$ be any subspace of $V$. Prove that there is a subspace $W_2$ of $V$ such that $V=W_1\oplus W_2$.…
Exercise 6.5.1 Find an invertible real matrix $P$ such that $P^{-1}AP$ and $P^{-1}BP$ are both diagonal, where $A$ and $B$ are the real matrices $$\text{(a)}\quad A=\left[\begin{array}{cc} 1&2\\0&2\end{array}\right],\quad B=\left[\begin{array}{cc} 3&-8\\0&-1\end{array}\right]$$$$\text{(b)} \quad…
Exercise 6.4.1 Let $T$ be the linear operator on $\mathbb R^2$, the matrix of which in the standard ordered basis is $$A=\left[\begin{array}{cc}1&-1\\2&2\end{array}\right].$$(a) Prove that the only subspaces of $\mathbb R^2$…
Exercise 6.3.1 Solution: The minimal polynomial for the identity operator is $x-1$. It annihilates the identity operator and the monic zero degree polynomial $p(x)=1$ does not, so it must be the…
Exercise 6.2.1 In each of the following cases, let $T$ be the linear operator on $\mathbb R^2$ which is represented by the matrix $A$ in the standard ordered basis for…
Exercise 5.2.1 Each of the following expressions defines a function $D$ on the set of $3\times3$ matrices over the field of real numbers. In which of these cases is $D$…