Mathematics

Linear Algebra (Matrix/Vector)

3brown1blue videos playlist

• basis vectors
  • think of vector as an arrow point on the map, with origin at [0,0]
  • think of basis vector i and j and each x and y coordinate on the map
• linear transformation
  • the linear transformation can be thought of the squishing or enlarging the nD dimensional space
    • it won't be linear if the origin moves or the spacing becomes unequal
  • we can put the "new" basis vector as column 1, 2,... of the transformation matrix. we use the dot product to apply the transformation
  • 2x3 transformation matrix means we are squishing from 3D to 2D coordinates.
  • the order of transformation is important -- "rotate then shear" is not the same as "shear then rotate"
  • but it is associative -- you can compute the product of transformations (called compositions) as long as they are in the right order
• determinant
  • the det is actually the area/volume changed by the basis vector of that matrix -- either expanding or squishing
  • this is why (later) when det is zero, we might not have an inverse matrix
• inverse matrix
  • think of doing the transformation in reverse, that is how we find the inverse matrix
  • when the det is zero, we effective lose some dimension, which means many possible inverses (I think)
• cross product
  • (example: 2 dimension). the cross product between 2 vectors is
    • a unit vector orthogonalto both input vectors (which direction depends on the right-hand order)
    • then we scale this unit vector by the determinant (area) of the 2 input vectors
• change of basis
  • let A is a vector in basis₁, and B is transformation matrix in basis₂.
  • we can construct a change of basis vector Q for basis₁ --> basis₂
  • then we can make the transformation matrix C = inv_Q * B * Q
  • then we can apply to A, getting A_hat = C * A in basis₁
• eigen vectors and eigen values
  • eigen vectors are the "orthogonal" vectors that don't not change direction after some transformation Q
  • eigen values is the scaling value (lengthen/shorten) associated with certain eigen vectors
  • why do we need this? because we can use eigen vector to make "change of basis", then make transformation product much easier to calculate ( namely, scaling transformation only -- diagonal matrix)