Linear Algebra 2. Unit vectors, linear combinations, basis.

Each coordinate of a vector is a scalar stretching and squishing a unit vector. Unit vectors are vectors starting (as each vector) at the origin, orthogonal (perpendicular) to each other and having length of one unit on the corresponding axis. Unit can be anything you want - 1 centimeter, 1 meter, 1 millimeter etc. So:

1. unit vector on x axis is î (i-hat) with coordinates [1,0] meaning 1 of x, o of y
2. unit vector on y axis is ĵ (j-hat) with coordinates [0,1] meaning 0 of x, 1 of y
3. so first we write x coordinate, then y, then (if any) z etc.

So each and any vector is sum of scaled unit vectors. We use vector-scalar multiplication and then vector addition. Thus we make linear combination (as result we get a vector which is an arrow) of î and ĵ (here 3 and 2 are scalars):

[3,2] = 3î + 2 ĵ = 3[1,0] + 2[0,1] = [3,0] + [0,2] = [3,2]

î and ĵ also called basic vectors of the x-y coordinate system.

We can choose different basis vectors (non unit) and get completely new coordinate system. So when describing vectors numerically, it (description) depends on a choice of basis vectors.

These materials were used while preparing this blog-post:

1. https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
2. https://www.deeplearningbook.org/
3. NBGtLA by https://minireference.com/