# Tag Archives: Exercise C

## Chapter 8 Exercise C

## Chapter 7 Exercise C

2. Solution: Note that $T$ is a positive operator on $V$, we have \begin{equation}\label{7CP2.1} \langle T(v-w),v-w\rangle\ge 0. \end{equation}On the other hand, \[Tv=w\quad\text{ and } \quad Tw=v\]imply that $T(v-w)=w-v$, hence \begin{equation}\label{7CP2.2}\langle T(v-w),v-w\rangle=-\langle v-w,v-w\rangle\le 0.\end{equation}Therefore $\langle v-w,v-w\rangle=0$ by (\ref{7CP2.1}) and (\ref{7CP2.2}), i.e. … Continue reading

## Chapter 6 Exercise C

3. Solution: By 6.31, we have \[ \m{span}(e_1,\cdots e_m)=\m{span}(u_1,\cdots,u_m)=U. \]Note that $e_1,\cdots e_m$ is an orthonormal list, it follows that $e_1,\cdots e_m$, is an orthonormal basis of $U$. By 6.47, we have $V=U\oplus U^{\perp}$. As $e_1,\cdots e_m,f_1,\cdots,f_n$ is an orthonormal … Continue reading

## Chapter 5 Exercise C

1. Solution: It is not said $V$ is finite-dimensional, but I will do it by assuming $\dim V<\infty$. If $T$ is invertible, then $\m{null}{T}=0$ and $\m{range} T=V$ since $T$ is bijective and surjective. Hence $V=\m{null} T \oplus\m{range} T$. If $T$ … Continue reading

## Chapter 3 Exercise C

1. Suppose $V$ and $W$ are finite-dimensional and $T\in\ca L(V,W)$. Show that with respect to each choice of bases of $V$ and $W$, the matrix of $T$ has at least $\dim \m{range} T$ nonzero entries. Solution: Suppose for some basis … Continue reading

## Chapter 2 Exercise C

1. Solution: Let $u_1,u_2,\cdots,u_n$ be a basis of $U$. Thus $n=\dim U=\dim V$. Hence $u_1,u_2,\cdots,u_n$ is a linearly independent list of vectors in V with length $\dim V$. By 2.39, $u_1,u_2,\cdots,u_n$ is a basis of $V$. In particular, any vector … Continue reading

## Chapter 1 Exercise C

1. Solution: (a) $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=0\}$ is a subspace of $\mathbb F^3$. By 1.34, to show a subset is a subspace, we just need to check Additive identity, Closed under addition and Closed under scalar multiplication. Additive identity: it is clear … Continue reading