## 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,…

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,…

1. Find all vector spaces that have exactly one basis. Solution: The only vector spaces is $\{0\}$. For if there is a nonzero vector $v$ in a basis, then we…

1. Suppose $v_1$, $v_2$, $v_3$, $v_4$ spans $V$. Prove that the list\[v_1-v_2,v_2-v_3,v_3-v_4,v_4\] also spans $V$. Solution: We just need to show that $v_1$, $v_2$, $v_3$, $v_4$ can be expressed as…