227A-DIS1

10 flashcards · Mathematics · by John Viljoen

1. what is the rank nullity theorem?

   dim(null(A)) + dim(range(A)) = number of columns in A

2. let A have full column rank: Show that (A.T A) is invertible

   We know by rank nullity that if we can show A.T A has a zero null space, it must have full rank, and so must be invertible. -> A has full column rank -> columns ai are linearly independent -> Ax = 0 for x = 0 only -> A itself has the trivial nullspace of 0 -> we must demonstrate null(A) = null(A.T A) therefore null(A) = null(A.T A) = trivial nullspace. therefore A.T A is invertible.

3. prove null(A) = null(A.T A)

4. give 2 ways we can show that a matrix is invertible

   show null space is zero (trivial) -> through rank nullity its rank must be full, and therefore it must be invertible show that its determinant is non zero.

5. how can we show a determinant of A.T A is non zero for an arbritrary A of full rank? (recall this proves invertiblity)

   use identity: then use SVD: since A is full rank: therefore Sigma.T Sigma is therefore: and we know V is orthonormal, therefore: Therefore det(A.T A) != 0 therefore A.T A is invertible

9. what is another word for Gram-Schmidt Orthonormalisation (GSO)

   QR decomposition

10. let A have full rank, what is the form of its QR decomposition?

    where Q is an orthonormal matrix and R is an upper-triangular matrix. The columns of Q1 form an orthonormal basis for the range space of A, and the columns of Q2 form an orthonormal basis for the range space of A perpendicular: R1 is upper triangular and invertible