Notes from 3Blue1Brown video Vectors from the Essence of linear algebra series.

Vectors are the building block of linear algebra.

3 perspectives of vectors:

• Physics student
  • Vectors are arrows pointing in space defined by length & direction.
  • You can move a vector around and it remains the same vector.
  • If the space is a flat plane, it's a 2d vector.
• Computer scientist (or data scientist)
  • An orderer list of numbers.
  • Example, a house can be model as the vector: [square footage, sale price]
• Mathematician
  • A vector can be anything that has a notion of adding two together and multiplying a vector by a number.
• In Linear Algebra, the vector is nearly always rooted at the Origin, which is the place where x and y intersect on the coordinate system.
  • Note how it differs from physics student's perspective that can consider a vector at any point in space.
• The "back and forth" between the physics and computer science student understanding of vectors is that we represent coordinates for a line as lists of numbers.

• 2d coordinate system refresher

  • Has a horizontal line called the x-axis
  • Have a vertical line called the y-axis

  • Place where they intersect is called the Origin

    

    • Can be thought of as the center of space and root of all vectors.
      • An arbitrary length is chosen to represent one, then tick marks are created on each axis spaced at this distance

    

• Coordinates of vector give instructions to get from the Origin to its tip

  

  1. The first number describes how far to walk along the x-axis.
  2. The second number describes how far along the y-axis.

• We distinguish vectors from points by writing them vertically using square bracket notation.

• In 3d, an additional axis is drawn that's perpendicular to the other axes, adding a 3rd coordinate

  

• Vector addition from the physics perspective:

  1. First draw both vectors:

     

  2. Move the 2nd vector up to the tip of the tail of the first vector:

     

  3. Draw a new vector that starts from the tail of the first to the tip of the 2nd. The new vector represents the sum of 2 vectors.

     

• The above definition of vector addition is an extension to how we teach kids to add numbers using the number line: we start at a number, then step in the direction of the number we're adding.

  

• The same operation written from the computer scientist perspective:

  $\begin{bmatrix} 1 \\ 2 \end{bmatrix} + \begin{bmatrix} 3 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 + 3 \\ 2 + -1 \end{bmatrix} = \begin{bmatrix} 4 \\ 1\end{bmatrix}$

• Vector multiplication is when you multiply a vector by a number (Vector Scaling) either stretching or squishing the vector by that amount.

  

  • Referred to as "scaling the vector," hence the number in this context is called a "scalar."

  • Computer science perspective:

    $2 \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 2 * 1 \\ 2 * 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}$

Maths/LinearAlgebra/Vectors