# Week 1: Vectors and Matrices

## Dot Product

The dot product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is calculated as:
$$
\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta
$$
where $\theta$ is the angle between them.

### Example
Let $\mathbf{u} = (2, 3)$ and $\mathbf{v} = (4, 1)$:
$$
\mathbf{u} \cdot \mathbf{v} = 2(4) + 3(1) = 11
$$

---

## Magnitude and Direction

- **Magnitude:**
$$
\|\mathbf{v}\| = \sqrt{x^2 + y^2}
$$

- **Direction:**
$$
\theta = \arctan\left(\frac{y}{x}\right)
$$

---

## Trigonometric Definitions

### Polar and Cartesian Coordinates
- **From Cartesian to Polar:**
$$
r = \sqrt{x^2 + y^2}, \quad \theta = \arctan\left(\frac{y}{x}\right)
$$

- **From Polar to Cartesian:**
$$
x = r \cos \theta, \quad y = r \sin \theta
$$

---

## Solving Linear Equations

### Geometric Interpretation
Solving a system of linear equations involves finding the intersection of their corresponding lines (in 2D) or planes (in 3D).

### Matrix Example
For $\mathbf{A} \mathbf{x} = \mathbf{b}$, where:
$$
\mathbf{A} =
\begin{bmatrix}
2 & 1 \\
1 & -1
\end{bmatrix}, \quad
\mathbf{b} =
\begin{bmatrix}
3 \\
0
\end{bmatrix}
$$
Solve via elimination or substitution to find $\mathbf{x}$.

---

## Row Reduction

### Example: Gaussian Elimination
Given matrix:
$$
\mathbf{A} =
\begin{bmatrix}
1 & 2 & -1 \\
2 & -3 & 4 \\
-1 & 1 & 3
\end{bmatrix}
$$
Perform row reduction to reach row echelon form:
$$
\begin{bmatrix}
1 & 2 & -1 \\
0 & -7 & 6 \\
0 & 0 & -2
\end{bmatrix}
$$

---

## Linear Combinations

A **linear combination** involves scaling and summing vectors:
$$
c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots
$$

### Example
Let $\mathbf{v}_1 = (1, 0)$ and $\mathbf{v}_2 = (0, 1)$:
$$
3\mathbf{v}_1 + 2\mathbf{v}_2 = (3, 2)
$$

---

## Linear (In)dependence and Span

- **Linear Dependence:** Vectors are dependent if one can be written as a combination of others.
- **Span:** The set of all vectors that can be formed as linear combinations of a given set.

---

## Matrix Operations

### Matrix-Vector Multiplication
Matrix-vector multiplication applies a transformation.

### Example
$$
\mathbf{A} =
\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}, \quad
\mathbf{v} =
\begin{bmatrix}
5 \\
6
\end{bmatrix}
$$
$$
\mathbf{A} \mathbf{v} =
\begin{bmatrix}
1(5) + 2(6) \\
3(5) + 4(6)
\end{bmatrix} =
\begin{bmatrix}
17 \\
39
\end{bmatrix}
$$