I will update the homework after each lectures. It is due next Wednesday (since we have Labor day Monday)

1. Let �⊂�3V⊂R3 be the points that {(�1,�2,�3)∣�1+�2+�3=0}{(x1,x2,x3)∣x1+x2+x3=0}. Find a basis in $V$, and write the vector $(2,-1,-1)$ in that basis.

2. Let $V$ as above,. Let �=�2W=R2, let $V\to W$ be the map of forgetting coordinate �3x3. Is this an isomorphism? What's the inverse?

3. Let $V$ as above, and let$W$ be the line generated by vector $(1,2,3)$. Let $f:V\to W$ be the orthogonal projection, sending $v$ to the closest point on $W$. Is this a linear map? How do you show it? What's the kernel? Let $g:W\to V$ be the orthogonal projection. Is it a linear map? What's the relationship between $f$ and $g$?

4. about quotient space. Let �=�2V=R2, and let $W$ be the linear subspace generated by vector $(1,2)$ (i.e. the line passing through origin and $(1,2)$). For $v\in V$, let $[v]=v+W\in V/W$ denote the equivalence class that $v$ belongs to, i.e., the (affine) line parallel to $W$ and passing through $v$. Draw some pictures to answer these questions.

• Is it true that $[(0,0)]=[(1,2)]$?
• Is it true that $[(1,1)]=[(0,1)]$?

5. another important notion is dual vector space. Given a vector space $V$, the dual vector space is �∗=���(�,�)V∗=Hom(V,R), the set of linear maps from $V$ to $\mathbb{R}$ (Hom is short for 'homomorphism', which means linear maps for vector spaces). For example, if �=�2V=R2, the linear functions $x$ and $y$ belong to �∗V∗, we have �∗={��+��∣�,�∈�}V∗={ax+by∣a,b∈R}. Here $x,y$ are basis for �∗V∗.

Let $V$ be the vector space of polynomials with degree less or equal than 3. What's the dimension of $V$? What's the dimension of �∗V∗? Can you find a basis for $V$? A basis for �∗V∗?

1. Here is claim $1+2+3+4+\cdots =-1/12$. Show that this is wrong.

Fun fact: There is an interesting function , called Riemann Zeta function $\zeta (s)$, which for $s>1$ can be written as �(�)=∑�=1∞1/��ζ(s)=∑n=1∞1/ns. In fact $\zeta (s)$ is actually a meromorphic function of $s$, and $\zeta (-1)=-1/12$.

2. Does the following series converge? Explain why.

• ∑�=1∞1/�2∑n=1∞1/n2.
• ∑�=1∞1/�!∑n=1∞1/n!
• ∑�=1∞�2/�!∑n=1∞n2/n!

3. Let ��an be a sequence of $\pm 1$. Show that ∑�=1∞��/2�∑n=1∞an/2n is convergent. (Hint: absolute convergence implies convergence)

4. What is radius of convergence? Is it true that11−�=1+�+�2+⋯1−x1=1+x+x2+⋯holds for all real number $x\ne 1$?

5. We know that the following series diverge1+1/2+1/3+1/4⋯.1+1/2+1/3+1/4⋯.Question: does the following alternating series converge? Why?1−1/2+1/3−1/4+⋯1−1/2+1/3−1/4+⋯(Optional): Fix any real number $a$. Show that by rearrange the order of the terms in the above alternating series, we can have the series converges to $a$.

6. Line integral: let $\gamma$ be the straightline from $(0,0)$ to $(1,1)$. Compute the line integral∫�2��+3��.∫γ2dx+3dy.What if we replace $\gamma$ by a curved line but still from $(0,0)$ to $(1,1)$, would the above result change? Why?