Linear Algebra.md

Brilliant Linear Algebra Course

Vector Space

A vector space $V$ is a non-empty set of objects $|v\rangle$ with some kind of addition and scalar multiplication that satisfy the following:

• Additive associativity: $|u\rangle+(|v\rangle+|w\rangle)=(|u\rangle+|v\rangle)+|w\rangle$
• Multiplicative associativity: $a\left(b|v\rangle\right)=(ab)|v\rangle$
• Additive commutativity: $|v\rangle+|w\rangle=|w\rangle+|v\rangle$
• Existence of $|0\rangle:|v\rangle+|0\rangle=|v\rangle,|v\rangle+(-1)|v\rangle=|0\rangle$
• Multiplicative identity: $1|v\rangle=|v\rangle$ for all elements in $V$
• Scalar distributivity: $a\left(|v\rangle+|w\rangle\right)=a|v\rangle+a|w\rangle$
• Vector distributivity: $(a+b)|v\rangle=a|v\rangle+b|v\rangle$

Waves as Vectors

The notation from the last animation is called a ket. It's a “wrapper” used to indicate that an object is a vector by writing "object" as $| \text{"object"}\rangle$. Waves wrapped in ket notation to emphasize their “vector” nature.

Signals

A Signal is the process and result of transmitting data over time. Generally real-valued function $f(t)$ and the set of all these functions is $F$.

All waves $\in F$ but the are special since they obey the property $w(t+1) = w(t)$ we note this subset of $F$ as $W$ ($W \subset F$).

Both $F$ and $W$ are Vector Spaces This means you can call $W$ a Subspace of $F$.

TO approximate a signal we can ask. What $|w \rangle \in W$ is as close to $|f \rangle \in F$?

Cartesian Plane

The plane is made up of all real number pairs $(x, y)$ and addition and scalar multiplication are defined as follows $$(x_1, y_1) + (x_2, y_2) = (x_1+x_2, y_1+y_2)$$$$c(x, y) = (cx, cy)$$ This makes the Cartesian Plane a Vector Space

Lines

Lines that pass through the origin in the cartesian plane, is a Subspace of the Cartesian Plane

The Gauss-Jordan Process

A Linear System is a collection of one or more linear equations in the same set of variables $x_1, \dotsc, x_n$ in the form $$a_{i1}x_1+a_{i2}x_2 + \dots + a_{in}x_n=b_i$$
$$\left{\begin{array}{c}a_{11}x_1+\cdots+a_{1n}x_n=b_1\\vdots\a_{m1}x_1+\cdots+a_{mn}x_n=b_m\end{array}\right.\mapsto\left(\begin{array}{ccc|c}a_{11}&\dots&a_{1n}&b_1\\vdots&\vdots&\vdots&\vdots\a_{m1}&\dots&a_{mn}&b_m\end{array}\right)$$