Today we will continue our review of linear algebra. Hopefully you have brushed up on the set notations over the weekends.

Let $V$ be a vector space, and $W \In V$ be a subspace. The quotient space $V/W$ is the following vector space:

• as a set, it consist of element that looks like the shifted linear subspaces $[v] = v+W =\{v + w \mid w \in W\}$. We note that, if $v_1, v_2 \in V$ satisfies $v_1 - v_2 \in W$, then $[v_1]=[v_2]$.
• the vector space operations are defined using the representatives, namely
  • $ c [v]= [cv], \forall c \in \R, v \in V$
  • $[v_1] + [v_2] = [v_1 + v_2]$

Motivation: why we care about quotient space? What's the meaning of the subspace? When we quotient something out, we are defining some equivalence relation, and we are ignoring some differences. In the quotient vector space case, suppose we want to identify the vectors in space $W$ as $0$, we say two points $v_1, v_2$ are equivalent, if their difference is in $W$.

A basis in $V$ is a collection of vectors, such that they are maximally linearly independent.

Given a basis, we can express all other vectors using linear combination of the basis. The coefficiients in the linear combination are called coordinates.

An inner product on a vector space is a function $(,): V \times V \to \R$, such that

• symmetric: $(v,w) = (w,v)$
• linear in each slot: $(av_1 + bv_2, w) = a (v_1, w) + b(v_2, w)$ (by symmetry, also linear in the 2nd slot)
• positive definite. $(v,v) \geq 0$ and $=0$ only if $v=0$.

You may be very familiar with the notion of $\R^n$, equipped with a (default) Euclidean inner product. But in general, for a vector space $V$, the inner product is something that you give it afterwards.

A nice basis for vector space with inner product is called an orthonormal basis. $e_1, \cdots, e_n \in V$, such that $(e_i,e_j) = \delta_{ij}$.

If $V$ is a vector space with an inner product, and $W \In V$ is a subspace, then we can define some projection $$\pi: V \to W $$ it satisfies that $ v - \pi(v) \perp \pi(v). $

Let $V \In \R^3$ be the points that $\{(x_1, x_2, x_3) \mid x_1 + x_2 + x_3=0\}$. Find a basis in $V$, and write the vector $(2,-1,-1)$ in that basis.

Let $W = \R^2$, let $V \to W$ be the map of forgetting coordinate $x_3$. Is this an isomorphism? What's the inverse?

Let $V$ as above, and $W$ 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 relatino between $f$ and $g$?