This will be my summary for: “Introduction to Linear Algebra - Gilbert Strang”.

## 1. Vectors and Matrices

Linear algebra is about vectors and matrices. These are the basic objects that we can add, subtract and multiply (when there shapes match).

### 1.1 Vectors and Linear Combinations

Linear Algebra begins with vectors $v,w$ and their linear combinations $cv+dw$. This takes you to two or more dimensions.

Linear combinations use two basic operations: 1) Scalar multiplication Scaling and 2) Adding vectors Addition.

The concept of linear combinations opens up two key questions:

- Describe all the combinations. Which
__space__do they fill? - Find the values for the coefficients which produce a
__specific__combinations.

In $R_{2}$ we can answer these questions quite simply:

- $cv$ for all c fills an infinite line in the xy-plane. If w happens to be on that line, then combining linear combinations of v and w won’t “escape” that line. If w is independent of v then the entire xy-plane can be reached as a linear combination of v and w.
- Two equations and two variables → go solve them.

### 1.2 Lengths and Angles from Dot Products

#### Dot Products

One of the most useful multiplications of vectors is the dot product. The __dot product__ (aka inner product) of a two vectors $v⋅w$ is $v_{1}w_{1}+v_{2}w_{2}+⋯+v_{n}w_{n}$ (See Dot Product (Scalar Product))

The dot product $v⋅v$ tells us the __squared length__: $∣∣v∣∣_{2}=v_{1}+v_{2}+⋯+v_{n}$.

#### Length/Magnitude

The __length__ $∣∣v∣∣$ (aka magnitude) of a vector is $v_{1}+v_{2}+⋯+v_{n} $ (See Why Does Squaring and Taking the Square Root Give the Length?). A diagonal in n dimensions has the length $n $.

#### Unit Vector

We use the words __unit vector__ are used when the length of a vector = 1. Otherwise: $u=∣∣v∣∣v $ gives us the unit vector. (See: Unit Vectors)

For example, $i=(10 )$ and $u=(cosθsinθ )$. Notice that $u$ is a unit vector because $u⋅u=1$.

#### Perpendicular Vectors

Suppose the angle between v and w is 90°. It’s cosine is zero. $v⋅w=0$ for perpendicular vectors.

Then we can use the Pythagorean Theorem and deduce that $v⋅w$ must be 0. The formula is saying take the hypotenuse which is $v+w$. Now take the length of that and square it. This must be equal to the individual side lengths (vector magnitudes) squared summed up. Expanding the formula reveals that the $v⋅w$ terms all need to be zero.

**Cosine of the angle:**
Let us take the unit vectors $v=(1,0)$ and $u=(cosθ,sinθ)$ then $v⋅u=cosθ$. We’ll try to connect the dot product to the angle between vectors.

#### Angle Between Two Vectors

Let the unit vectors $v=(cosa,sina)$ and $w=(cosb,sinb)$ Then we have $v⋅w=cosacosb+sinasinb=cos(b−a)$ (Using trigonometric identities).