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## Solution to Linear Algebra Hoffman & Kunze Chapter 9.2

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…

## Solution to Linear Algebra Hoffman & Kunze Chapter 7.5

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…

## Solution to Linear Algebra Hoffman & Kunze Chapter 7.1

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,…

## Solution to Linear Algebra Hoffman & Kunze Chapter 6.8

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…

## Solution to Linear Algebra Hoffman & Kunze Chapter 6.6

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 $V=W_1\oplus W_2$.…

## Solution to Linear Algebra Hoffman & Kunze Chapter 6.3

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…

## Solution to Linear Algebra Hoffman & Kunze Chapter 6.2

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…

## Solution to Linear Algebra Hoffman & Kunze Chapter 5.2

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 $D$…