Math for Data Science Exam #2

Eigenvector/Eigenvalue

Let a be an nxn matrix, and let x be a non-zero vector such that Ax=λx for some scalar λ. Then x i an eigenvector of A with corresponding eigenvalue λ.

Rank =

Number of pivots

Nullity =

Number of free variables or Dimension

Dimension =

Number of free variables or Nullity

Rank + Nullity =

Number of Columns in the Matrix

Column Space

The span of the matrix. Find by doing rref, and only returning the columns from the original matrix that have a pivot in them. The dimension of the column space is the number of elements.

Projection of y onto the subspace generated by u

yhat = (y*u)/(u*u) * u

Projection of a point onto a line

projection = (y*d)/(d*d) * d; d = direction vector of the line

Best fit line

(A(transpose)*A)^-1= A(transpose)*b