Solution to Linear Algebra Hoffman & Kunze Chapter 4.5
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…
Solution to Linear Algebra Hoffman & Kunze Chapter 4.4
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…
Solution to Linear Algebra Hoffman & Kunze Chapter 4.3
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: $t_0=-1$, $t_1=0$,…
Solution to Linear Algebra Hoffman & Kunze Chapter 4.2
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…
Solution to Linear Algebra Hoffman & Kunze Chapter 3.7
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…
Solution to Linear Algebra Hoffman & Kunze Chapter 3.6
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 $W^0$…
Solution to Linear Algebra Hoffman & Kunze Chapter 3.5
Exercise 3.5.1 In $\mathbb R^3$ let $\alpha_1=(1,0,1)$, $\alpha_2=(0,1,-2)$, $\alpha_3=(-1,-1,0)$. (a) If $f$ is a linear functional on $\mathbb R^3$ such that$$f(\alpha_1)=1,\quad f(\alpha_2)=-1,\quad f(\alpha_3)=3,$$and if $\alpha=(a,b,c)$, find $f(\alpha)$. (b) Describe explicitly…
Solution to Linear Algebra Hoffman & Kunze Chapter 3.4
Exercise 3.4.1 Let $T$ be the linear operator on $\mathbb C^2$ defined by $T(x_1,x_2)=(x_1,0)$. Let $\mathcal B$ be the standard ordered basis for $\mathbb C^2$ and let $\mathcal B'=\{\alpha_1,\alpha_2\}$ be…
Solution to Linear Algebra Hoffman & Kunze Chapter 3.3
Exercise 3.3.1 Let $V$ be the set of complex numbers and let $F$ be the field of real numbers. With the usual operations, $V$ is a vector space over $F$.…
Solution to Linear Algebra Hoffman & Kunze Chapter 3.2
Exercise 3.2.1 (a) Geometrically, in the $x-y$ plane, $T$ is the reflection about the diagonal $x=y$ and $U$ is a projection onto the $x$-axis. (b) We have $(U+T)(x_1,x_2)=(x_2,x_1)+(x_1,0)=(x_1+x_2,x_1)$. $(UT)(x_1,x_2)=U(x_2,x_1)=(x_2,0)$. $(TU)(x_1,x_2)=T(x_1,0)=(0,x_1)$.…