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 $D$…
Exercise 4.5.1 Let $p$ be a monic polynomial over the field $F$, and let $f$ and $g$ be relatively prime polynomials ovef $F$. Prove that the g.c.d. of $pf$ and…
Exercise 4.4.1 Let $\mathbb Q$ be the field of rational numbers. Determine which of the following subsets of $\mathbb Q[x]$ are ideals. When the set is an ideal, find its…
Exercise 4.3.1 Use the Lagrange interpolation formula to find a polynomial $f$ with real coefficients such that $f$ has degree $\leq 3$ and $f(-1)=-6$, $f(0)=2$, $f(1)=-2$, $f(2)=6$. Solution: $t_0=-1$, $t_1=0$,…
Exercise 4.2.1 Let $F$ be a subfield of the complex numbers and let $A$ be the following $2\times 2$ matrix over $F$ $$A=\left[\begin{array}{cc}2&1\\-1&3\end{array}\right].$$For each of the following polynomials $f$ over…
Exercise 3.7.1 Let $F$ be a field and let $f$ be the linear functional on $F^2$ defined by $f(x_1,x_2)=ax_1+bx_2$. For each of the following linear operators $T$, let $g=f^tf$, and…
Exercise 3.6.1 Let $n$ be a positive integer and $F$ a field. Let $W$ be the set of all vectors $(x_1,\dots,x_n)$ in $F^n$ such that $x_1+\cdots+x_n=0$. (a) Prove that $W^0$…