1. Define addition and scalar multiplication on the set $\{(a_1,a_2,\ldots) a_i\in\mathbb{R}\}$ to prove that it forms a vector space.

  2. Let’s define a differential operation $D:V\to V$ as

    \[D(\{a_n\})=\{a_{n+1}-a_{n}|n=1,2,3,\ldots\}.\]

    Show that the set of solutions of $D^2-2D+3=\{0\}$ forms a subspace of the above vector space, where $\{0\}$ is the sequence of $0$. We define $D^2(\{a_n\})=D(D(\{a_n\}))$ and $(D^2-2D+3)(\{a_n\})=D^2(\{a_n\})-2\cdot D(\{a_n\})+3\cdot\{a_n\}.$

  3. Prove that $\{(a_1,a_2,\ldots)\}$ has infinite dimension. Then use this result to deduce that the set of all real functions defined on $\mathbb{R}$ also has infinite dimension.
  4. If a vector space $V$ is generated by a finite set $S_0$, then a subset of $S_0$ is a basis for $V$. Hence $V$ has a finite basis.
  5. Develop a strategy to address the following problem: Given a finite set $S$ in a vector space $V$, which may generate either $V$ or a subspace of $V$, find a subset of $S$ that forms a basis for $\text{span}(S)$.
  6. Use the strategy you developed above to find the subset in the set $\{(2,-3,5,0),(8,-12,20,2),(1,0,-2,4),(0,2,-1,3),(7,2,0,1)\}$

  7. For the following questions, prove that $T$ is a linear transformation and find bases for both $N(T)$ and $R(T)$. Then compute the nullity and rank of $T$ and verify the equality

    \[\text{nullity}(T)+\text{rank}(T)=\text{dim}(T)\]

    and determine whether $T$ is one-to-one or onto.

    1. $T:P_2(\mathbb{R})\to P_3(\mathbb{R})$; $T(f(x))=xf(x)+f’(x).$

    2. $T:M_{n\times n}(\mathbb{R})\to \mathbb{R}$; $T(A)=\text{Trace}(A)$. Recall that

      \[\text{Trace}(A)=\sum_{i=1}^n a_{ii}\]

      where $A=(a_{ij})$.