lecture 4: systems of linear equations Flashcards
definition of matrix inverse
a square matrix A is said to be invertible if there is a square matrix such that AB = BA = I (identity matrix)
* A, B and I are all the same size
the adjugate/adjoint of matrix A is
the transpose of the cofactor matrix
the (i,j) entry of cofactor matrix is
the (i,j) minor times a sign factor
a collection of d-vectors(vectors with d features) is linearly independent if
the sum of all the vectors cannot equal zero unless all of the coefficients are zero
a system of linear equations can be denoted in matrix vector form using
Xw = y
an even determined system has
an equal number of equations and unknowns
an over-determined system has
more equations than unknowns, which means it has no exact solution
an under-determined system has
more unknowns than equations, which means it has an infinite number of solutions
how to solve an over-determined system
an approximate solution can be obtained using left inverse
w = (XᵀX)⁻¹Xᵀy
(XᵀX)⁻¹Xᵀ = left inverse
how to solve an under-determined system
unique solution possible by using right inverse
w = Xᵀ(XXᵀ)⁻¹y
Xᵀ(XXᵀ)⁻¹ = right inverse
