Tommy Deng Notes

Introduction to Linear Algebra

Vectors are n-tuples of numbers.

x⃗=[x1x2⋮xn]xi∈R \vec{x} = \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix} \quad x_i \in\mathbb{R}

Different notations exist, such as x⃗=(x1,x2,…,xn)\vec{x} = (x_1, x_2, \ldots, x_n)

Operations With Vectors

Equality

x⃗=[x1x2⋮xn]xi∈R\vec{x} = \begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix} \quad x_i \in \mathbb{R} and y⃗=[y1y2⋮yn]yi∈R\vec{y} = \begin{bmatrix}y_1\\y_2\\\vdots\\y_n\end{bmatrix} \quad y_i \in \mathbb{R}

are equal if each corresponding value is the same.

x1=y1,x1=y1,…,xn=ynx_1 = y_1, x_1 = y_1, \ldots, x_n = y_n

Addition and Subtraction

Combining vectors (recall "tip to tail").

[x1x2⋮xn]±[y1y2⋮yn]=[x1±y1x2±y2⋮xn±yn]\begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix} \pm \begin{bmatrix}y_1\\y_2\\\vdots\\y_n\end{bmatrix} = \begin{bmatrix}x_1 \pm y_1\\x_2 \pm y_2\\\vdots\\x_n \pm y_n\end{bmatrix}

Scalar Multiplication

Scaling done on the vector which only changes magnitude. A negative refers to 'flipping' or reversing its direction.

cx⃗=[cx1cx2⋮cxn]c∈Rc\vec{x} = \begin{bmatrix}cx_1\\cx_2\\\vdots\\cx_n\end{bmatrix} \quad c\in\mathbb{R}

Dot Product

Directional multiplication.

[x1x2⋮xn]⋅[y1y2⋮yn]=x1y1+x2y2+…+xnyn\begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix} \cdot \begin{bmatrix}y_1\\y_2\\\vdots\\y_n\end{bmatrix} = x_1y_1 + x_2y_2 + \ldots + x_ny_n

See Better Explained

Magnitude (Length)

Essentially an n-dimensional pythagorean theorem to find the length of a vector.

∥x⃗∥=x12+x22+…+xn2\|\vec{x}\| = \sqrt{x_{\tiny 1}^2 + x_{\tiny 2}^2 + \ldots + x_{\tiny n}^2}

Orthogonality (Perpendicular)

x⃗⊥y⃗⇔x⃗⋅y⃗=0\vec{x}\bot\vec{y} \Leftrightarrow \vec{x}\cdot\vec{y} = 0

Angles

cos(θ)=x⃗⋅y⃗∥x⃗∥∥y⃗∥cos(\theta) = \frac{\vec{x}\cdot\vec{y}}{\|\vec{x}\|\|\vec{y}\|}

For the acute angle, find θ\theta where 0≤θ≤π0\le\theta\le\pi.

Two vectors are parallel if θ=0\theta = 0 or θ=π\theta = \pi.

Projections

proju⃗v⃗=u⃗⋅v⃗∥u⃗∥2u⃗proj_{\vec{u}}\vec{v} = \frac{\vec{u}\cdot\vec{v}}{\|\vec{u}\|^2}\vec{u}